My main takeaway from the NYT profile is that I can't believe anyone ever claimed he wasn't a reactionary
oh and that the structuralists refuted his whole theory 60 years ago
Actual reactionaries, especially the meaner ones, think Peterson is a fairly boring pro-liberalism conservative (ie right-center/center-right) and they would be mostly correct on that. The idea that because he doesn't like modern feminism or pronouns that begin with xyz, he's actually aligned with people who want to overturn democracy or global capitalism is silly. When he isn't the middle of lecturing on his Jungian/Darwinian/Gnostic blend (he's a little less Christian than I originally thought, but in a way that makes him even less of a reactionary), he's a fairly standard knee-jerking Canadian conservative boomer, right down to failing to properly label what he's arguing against ("post-modern marxism", which is a misnomer for the typical "PC Culture" academia or maybe even Harold Bloom's School of Resentment). Though a statement like this is pointless if said to someone who thinks a person like (the much more straightforward, materialist) Pinker is an out and out reactionary (which I have seen on "post-modern marxist" style blogs) because he also relies on ideas of evolutionary biology to argue against the blank slate. Despite having very different attitudes, Pinker and Peterson are more or less arguing for the same kind of society with occasionally similar arguments, but Peterson wants that gooey center to be Judaeo-Chrstian & Dragons spiritual, while Pinker's soulless appetite is sated only with charts that keep pointing up. The fact the charts are already going up is a bit of a rhetorical obstacle for anyone who thinks there is a huge economical/political problem with society, left or right.
As for Peterson telling people to get married, my impression from reading the internet fallout of various attempts to handle the
Incel Question, it would be much more palatable to perform society-wide conversion therapy to make virginal men gay/women/get daily prostate milkings from prostitutes/etc (this seems not only increasingly possible, but might even already be happening judging by some of the nintendo/weeaboo/"redeemed through social justice"ex-4channer twitter accounts I constantly run into) than go back to marrying in your 20s and staying married into your 60s with a stable nuclear family structure. I don't think you can pin this all on women though, since men are probably not going to give up their 4kHD porn addictions anytime soon. History, from Viking raids to Chinese revolts, agrees with Peterson that it is risky to disregard such a powerful social technology like monogamy, but those barbaric peoples of the past never had SSRIs or the Marvel Cinematic Universe.