It is in the nature of the beast to publish such things under such headlines. New York Times gotta New York Times: 
First thought: “What about the over 100 million people who died because of this man’s philosophy?! What is wrong with you people?!”
Second thought, after reading the thing: “The philosopher has a point, but not really the one he thinks he has.”
Barker points out that Marx was correct that “capitalism has an inbuilt tendency to destroy itself.” I would add that Marx’s view that capitalism was heretofore the most revolutionary force in human history is also true. From the Communist Manifesto :
The bourgeoisie, historically, has played a most revolutionary part.
The bourgeoisie, wherever it has got the upper hand, has put an end to all feudal, patriarchal, idyllic relations. It has pitilessly torn asunder the motley feudal ties that bound man to his “natural superiors”, and has left remaining no other nexus between man and man than naked self-interest, than callous “cash payment”. It has drowned the most heavenly ecstasies of religious fervour, of chivalrous enthusiasm, of philistine sentimentalism, in the icy water of egotistical calculation. It has resolved personal worth into exchange value, and in place of the numberless indefeasible chartered freedoms, has set up that single, unconscionable freedom — Free Trade. In one word, for exploitation, veiled by religious and political illusions, it has substituted naked, shameless, direct, brutal exploitation.
The bourgeoisie has stripped of its halo every occupation hitherto honoured and looked up to with reverent awe. It has converted the physician, the lawyer, the priest, the poet, the man of science, into its paid wage labourers.
The bourgeoisie has torn away from the family its sentimental veil, and has reduced the family relation to a mere money relation.
The bourgeoisie has disclosed how it came to pass that the brutal display of vigour in the Middle Ages, which reactionaries so much admire, found its fitting complement in the most slothful indolence. It has been the first to show what man’s activity can bring about. It has accomplished wonders far surpassing Egyptian pyramids, Roman aqueducts, and Gothic cathedrals; it has conducted expeditions that put in the shade all former Exoduses of nations and crusades.
The bourgeoisie cannot exist without constantly revolutionising the instruments of production, and thereby the relations of production, and with them the whole relations of society. Conservation of the old modes of production in unaltered form, was, on the contrary, the first condition of existence for all earlier industrial classes. Constant revolutionising of production, uninterrupted disturbance of all social conditions, everlasting uncertainty and agitation distinguish the bourgeois epoch from all earlier ones. All fixed, fast-frozen relations, with their train of ancient and venerable prejudices and opinions, are swept away, all new-formed ones become antiquated before they can ossify. All that is solid melts into air, all that is holy is profaned, and man is at last compelled to face with sober senses his real conditions of life, and his relations with his kind.
The need of a constantly expanding market for its products chases the bourgeoisie over the entire surface of the globe. It must nestle everywhere, settle everywhere, establish connexions everywhere.
You see what he means here. Capitalism — for Marx, the merchant class (the “bourgeoisie”) were the carriers of capitalism — turns everything into a market. Capitalism is a revolutionary force that disrupts and desacralizes all things. All that talk in The Benedict Option  about “liquid modernity”? That’s based in Marx, actually. Zygmunt Bauman, the late sociologist from whom I took the idea, was a Marxist.
Look, most of us conservatives in the West are to some degree supporters of the free market. What we missed for a very long time was that it is hard to support a fully free market while at the same time expecting our social institutions — the family, the church, and so forth — to remain stable. This is an insight of Marx’s that we conservatives — and even conservative Christians — ought to absorb. I write about this a lot, though not in specific Marxist terms.
The thing is, Christian Democratic parties throughout Western Europe have largely absorbed this truth. Catholic social teaching is based in these insights as well. They aren’t necessarily against the free market, but rather say that the market must be tempered for the common good.
That wasn’t Marx’s view, obviously. Marx thought the free market was itself wicked, and ought to be totally controlled by the state. We know where that all ended up: with a hundred million dead, and entire economies and societies destroyed.
But we can agree that Marx was right to diagnose the revolutionary nature of capitalism, if catastrophically wrong about the cure for capitalism’s excesses. If that was as far as Jason Barker went, that would be fine. But he doesn’t — and this is the warning. Barker continues:
The key factor in Marx’s intellectual legacy in our present-day society is not “philosophy” but “critique,” or what he described in 1843 as “the ruthless criticism of all that exists: ruthless both in the sense of not being afraid of the results it arrives at and in the sense of being just as little afraid of conflict with the powers that be.” “The philosophers have only interpreted the world, in various ways; the point is to change it,” he wrote in 1845.
Racial and sexual oppression have been added to the dynamic of class exploitation. Social justice movements like Black Lives Matter and #MeToo, owe something of an unspoken debt to Marx through their unapologetic targeting of the “eternal truths” of our age. Such movements recognize, as did Marx, that the ideas that rule every society are those of its ruling class and that overturning those ideas is fundamental to true revolutionary progress.
We have become used to the go-getting mantra that to effect social change we first have to change ourselves. But enlightened or rational thinking is not enough, since the norms of thinking are already skewed by the structures of male privilege and social hierarchy, even down to the language we use. Changing those norms entails changing the very foundations of society.
There it is, reader. There is the “cultural Marxism” that you hear so much about, and that so many on the left deny. It is in the Marxist principle that there is no such thing as truth; there is only power.
Lenin understood this well. This is the meaning of his famous dictum, “Who, whom?”  In Lenin’s view, co-existence with capitalism was not possible. The only question was whether or not the communists will smash the capitalists first, or the other way around. One way of interpreting this is to say that the moral value of an action depends on who is doing it to whom.
This is why it is pointless for us conservatives and old-school liberals to stand around identifying contradictions and hypocrisies in how the progressives behave. They don’t care! They aren’t trying to apply universal standards of justice. They believe that “universal standards of justice” is a cant phrase to disguise white heterosexist patriarchal supremacy. They believe that justice is achieving power for their group, and therefore disempowering other groups. This is why it’s not racist, in their view, to favor non-whites over whites in the distribution of power. This is why they don’t consider it unfair to discriminate against men, heterosexuals, and other out-groups.
They will use things like “dialogue” as a tactic to serve the long-term strategy of acquiring total power. Resisting them on liberal grounds is like bringing a knife to a gun fight. The neoreactionaries have seen this clearly, while conservatives like me, who can’t quite let go of old-fashioned liberalism, have resisted it.
I have resisted it because I really would like to live in a world where we can negotiate our differences while allowing individuals and groups maximum autonomy in the private sphere. I want to be left alone, and want to leave others alone. This, I fear, is a pipe dream. Absent a shared cultural ethos, I can’t see how this is possible. I hate to say it — seriously, I do — but I think that today’s conservatives (including me) are going to end up as neoreactionaries, just as today’s old-school liberals are going to end up as progressives, because the forces pulling us to these extremes are stronger than any centrism.
For example, check this out:
I’m running into irreligious people who think that a religious person violating their deeply held principles is just a matter of choice, that they don’t truly have any genuine beliefs.
We can’t even converse any more b/c we’re not speaking the same language.
— PoliticalMath (@politicalmath) May 1, 2018 
This is our country — and this is the danger we religious people are facing, and are going to face much more intensely. Many non-religious people simply cannot understand why we see the world the way we do, and assume that it can only be out of irrationality and bigotry.
I invite you to read this blog post from three years ago,  based on my interview with “Prof. Kingsfield”, a closeted Christian teaching at an elite law school. This excerpt:
“Alasdair Macintyre is right,” he said. “It’s like a nuclear bomb went off, but in slow motion.” What he meant by this is that our culture has lost the ability to reason together, because too many of us want and believe radically incompatible things.
But only one side has the power. When I asked Kingsfield what most people outside elite legal and academic circles don’t understand about the way elites think, he said “there’s this radical incomprehension of religion.”
“They think religion is all about being happy-clappy and nice, or should be, so they don’t see any legitimate grounds for the clash,” he said. “They make so many errors, but they don’t want to listen.”
To elites in his circles, Kingsfield continued, “at best religion is something consenting adult should do behind closed doors. They don’t really understand that there’s a link between Sister Helen Prejean’s faith and the work she does on the death penalty. There’s a lot of looking down on flyover country, one middle America.
“The sad thing,” he said, “is that the old ways of aspiring to truth, seeing all knowledge as part of learning about the nature of reality, they don’t hold. It’s all about power. They’ve got cultural power, and think they should use it for good, but their idea of good is not anchored in anything. They’ve got a lot of power in courts and in politics and in education. Their job is to challenge people to think critically, but thinking critically means thinking like them. They really do think that they know so much more than anybody did before, and there is no point in listening to anybody else, because they have all the answers, and believe that they are good.”
This is a small part of a larger struggle.
Many on the left deny that cultural Marxism exists, but you have in The New York Times a column by a Marxist professor saying that yes it does, and it’s a good thing, too. His final line:
On that basis, we are destined to keep citing him and testing his ideas until the kind of society that he struggled to bring about, and that increasing numbers of us now desire, is finally realized.
Marx didn’t come from nowhere. The world of 1848 (when the Communist Manifesto appeared) is a lot like our own world; re-read the section above from that document and see how familiar it sounds. He was more or less right in his diagnosis of the revolutionary nature of capitalism, but his materialism and its relationship to human nature was catastrophically wrong. His thought may have resulted in mass murder, but it is clearly not dead; it is simply turned against culture, not the means of production.
Therefore, I’ll end here with this excerpt from Carlo Lancellotti’s recent Commonweal essay about Marx, culture, and Catholicism. Excerpt:
Contra the “Catholic Left,” which tended to regard Marx’s atheism as accidental, and tried to rescue his socio-political analysis from his religious views, Del Noce concluded that what Marx proposed was not just a new theory of history or a new program of political economy, but a new anthropology, one completely different from the Christian tradition. (Louis Dupré had made a similar argument in the pages of Commonweal; see “Marx and Religion: An Impossible Marriage,”  April 26, 1968.) Marx viewed humans as “social beings” entirely determined by historical and material circumstances rather than by their relationship with God. He viewed human reason as purely instrumental—a tool of production and social organization rather than the capacity to contemplate the truth and participate in the divine wisdom. Finally, Marx viewed liberation as the fruit of political action, not as a personal process of conversion aided by grace. Marxist politics was not guided by fixed and absolute ethical principles, because ethics, along with philosophy, was absorbed into politics. Del Noce concluded that there was no way to rescue Marx’s politics from his atheism, which had as much to do with his view of man as with his view of God.
Nonetheless, after World War II Marxism experienced a resurgence in Western Europe, not only among intellectuals and politicians but also in mainstream culture. But Del Noce noticed that at the same time society was moving in a very different direction from what Marx had predicted: capitalism kept expanding, people were eagerly embracing consumerism, and the prospect of a Communist revolution seemed more and more remote. To Del Noce, this simultaneous success anddefeat of Marxism pointed to a deep contradiction. On the one hand, Marx had taught historical materialism, the doctrine that metaphysical and ethical ideas are just ideological covers for economic and political interests. On the other hand, he had prophesied that the expansion of capitalism would inevitably lead to revolution, followed by the “new man,” the “classless society,” the “reign of freedom.” But what if the revolution did not arrive, if the “new man” never materialized?
In that case, Del Noce realized, Marxist historical materialism would degenerate into a form of radical relativism—into the idea that philosophical and moral concepts are just reflections of historical and economic circumstances and have no permanent validity. This would have to include the concept of injustice, without which a critique of capitalism would be hard, if not impossible, to uphold. A post-Marxist culture—one that kept Marx’s radical materialism and denial of religious transcendence, while dispensing with his confident predictions about the self-destruction of capitalism—would naturally tend to be radically bourgeois. By that, Del Noce meant a society that views “everything as an object of trade” and “as an instrument” to be used in the pursuit of individualized “well-being.” Such bourgeois society would be highly individualistic, because it could not recognize any cultural or religious “common good.” In the Communist Manifesto, Marx and Engels described the power of the bourgeois worldview to dissolve all cultural and religious allegiances into a universal market. Now, ironically, Marxist ideas (which Del Noce viewed as a much larger and more influential phenomenon than political Marxism in a strict sense) had helped bring that process to completion. At a conference in Rome in 1968, Del Noce looked back at recent history and concluded that the post-Marxist culture would be “a society that accepts all of Marxism’s negations against contemplative thought, religion, and metaphysics; that accepts, therefore, the Marxist reduction of ideas to instruments of production. But which, on the other hand, rejects the revolutionary-messianic aspects of Marxism, and thus all the religious elements that remain within the revolutionary idea. In this regard, it truly represents the bourgeois spirit in its pure state, the bourgeois spirit triumphant over its two traditional adversaries, transcendent religion and revolutionary thought.”
If Del Noce is correct, we may not have to worry about the cultural Marxists of our time taking total power, as consumer capitalism and its comforts will compromise their revolutionary spirit. When and if university presidents start kicking these bumptious brats out of college, the revolution will sputter like Occupy Wall Street did. But before it’s all over, they may end up destroying the institutions and ways of life that make life stable and meaningful. Then again, unrestrained capitalism has done the same thing. The problem with Marxism is that it burns the boats so that nobody can return, and calls the resulting fire enlightenment.
The warning is twofold: First, that cultural Marxism is a real thing willing and capable of doing real damage, and that you cannot negotiate with these people; and second, that unless capitalists figure out how to ameliorate the excesses of market and technological change on society, they are tempting fate, just as their 19th and early 20th century forebears did.
UPDATE: Reader Dave:
The bigger problem with the NYT piece that you either missed or didn’t feel added to your thesis is the irony that Marx’s critiques are seen as a good and carrying that forward cultural Marxist critiques are good, unless you are critiquing those critiques. You aren’t allowed to critique arguments from BLM or La Raza or LGBTQXYZ groups or etc because taking a critical eye to those groups is just hateful bigoted nonsense. Never mind that those groups’ manifestos generally don’t hold up to scrutiny, just accept it as a means to an end (even if that end isn’t really where we should like to be). In a world where there is no objective truth and all individuals’ “truths” are valid there is no basis culture or society. But you can’t bring that up, lest you be labeled an insensitive bigot who should be burned at the stake. My guess is if Marx were revived today he would be ashamed more of the intellectual rot his philosophy has spawned than he would over the millions of innocents dead.