I think the intellectual dark web traces its origins to the populist response to the 9-11 attacks and their aftermath. The main issue at the time was the inability of Western elites to recognize the threats from Mahometanism. I think in part this was a consequence of the end of the Cold War, in which Mahometan states which had become clients of the Soviet Union were released from obligations imposed by membership in the Soviet sphere and began freelancing into military adventurism and terrorist activity. The end of the Cold War was also unforeseen by the CIA and George XLI; the policy establishment was left without a coherent strategy. This probably fed into his electoral defeat in 1992.
Prior to the rise of the intellectual dark web (this article makes a stab at defining it, but I definitely exclude Ben Shapiro, who is an old-fashioned neo-conservative, and I would include others like Carl Benjamin -- but remember, this is the New York Times) there were two strains of US academic and policy thought, neo-liberalism and neo-conservatism. Both, as the election of 2016 showed, had proven themselves incapable of addressing the challenges represented by the 9-11 attacks and subsequent developments. The US military response, a neo-conservative policy, proved extravagantly expensive yet inconclusive.
Neo-liberal tolerance of mass Mahometan migration into Western societies proved self-destructive, but it was only one result of the social disorder fostered by neo-liberal domestic policies. Dissatisfaction with both strains of thought led in part to Donald Trump's electoral success in 2016. However, there has never been a coherent strain of thought behind the populism that's replacing the older and outdated philosophies -- no National Review has emerged to give any sort of systematic presentation to the views of people like Peterson or Benjamin, while Buckley's magazine is no longer much other than self-parody.
The problem is that neither Peterson, a fully credentialed product of the academic establishment, nor Benjamin, who appears to have an elite university education of some sort, has enough of a philosophical background to articulate a fully coherent philosophy. Benjamin is a professed atheist, while Peterson clearly endorses Darwinian selection as part of an essentially materialist world view. The problem for both is where they can find an origin for questions of virtue or truth, which both clearly find important.
Here is where Bp Barron's observations are productive. Sam Harris, who is also a member of the intellectual dark web, is a hard atheist. Peterson, not quite so hard -- as Bp Barron correctly points out, Peterson finds a kind of Jungian archetypal value in scripture that promotes some sort of transcendental virtue, as opposed to Harris, who sees scripture as an evolutionary vestige and religion as pure superstition.
Bp Barron then moves to a much more important point: Peterson is coming from a modernist philosophical background, which he expresses fairly clearly. Nobody would have much disagreed with any of his points 50 years ago, when I was an undergraduate. I picked up Peterson's world view fairly well at that time, now that I think about it. The problem is that it isn't consistent, and Descartes, Hume, and Kant, not to mention Hegel or Darwin, don't form an adequate philosophical foundation for moving forward.
Actually, one reason I'm a big fan of the Fringe TV series is how much of it is a sendup of modernist thought as it's crept into elite popular culture. My wife and I are near-cultic followers. One of our favorite episodes is Momentum Deferred, in which shapeshifters from the alternate universe hijack a shipment of cryonically frozen heads, which have been preserved by transhumanists who hope to reattach them to new bodies at some future date. The shapeshifters are interested in only one of the heads, and as they go through identifying each to find the one they want, they toss the rejects away down a ravine. From a Cartesian-transhumanist point of view, this is a hilarious blasphemy.
I think the collapse of modernist thought is part of what's behind the rise of the newer, so far inchoate, movement represented by Peterson. Bp Barron doesn't quite come out and say this, but I think he agrees with Prof Feser that Sam Harris and the other New Atheists aren't worth debating. Peterson, who does think clearly within the limitations of his academic background, is a different matter, as is, I think, Carl Benjamin.
In fact, Bp Barron's explication of Kant's views on reason is quite good. Bp Barron is a Thomist. A debate between him and Jordan Peterson would be worthwhile. The problem is that, with the decline of both neo-liberal and neo-conservative intellectual forums, I'm not sure in what context it could take place.