It seems to me that Haidt wants to take a self-critical poke at a highly liberal academy even as he self-consciously writes from within it. I think he sees value in some of the things conservatives have traditionally valued, and mourns (at least in part) the marginalization of many conservative values in so much of the intellectual world. He sees it as the yin losing its yang and taking things to a place seriously out of balance.
Perhaps when we stop and take note of how all these lines get so blurred in each of us as individuals, we can see that liberals and conservatives really aren't so different. Haidt has provocatively claimed that the liberal call for diversity fades to silence on the matter of thought. Suddenly authority (as in the authority of scientific consensus and expertise) really does matter and how dare anyone think differently on that! That specific diversity cannot be tolerated; and the liberal momentarily forgets his liberalism.
Not to be outdone, the evolution-hating conservative takes leave of his normal pension for authority and asks "what high authority?" How dare you ask me to respect your scientific authority -- I am entitled to my personal choice just like everybody else; I choose the authorities I want, the news outlets I want, the facts and reality I want, the biblical understanding I prefer, the gender I want to be ... it's all about choice now is it not? Such is his perceptive taunt to a liberal audience he knows quite well. And the conservative forgets (if only for a moment) his conservatism.
But here's the thing: of the two curious "amnesias" I describe above, I do believe it is the conservative's temporary lapse that is the most fleeting for him -- he engages in it the most knowingly and playfully; never really forgetting that he is, underneath all the play, still anchored in the whole concept of authority. The liberal's lapse seems to be the more enduring condition, and the liberal clings to his dogma with the greater tenacity, internalizing it and even managing to forget that he has it. It is cordoned off by a sacred veil that has yet to be torn.
And if you doubt this comparison of relative persistence, thinking that it is conservatives who have all the power right now, then engage in this little thought experiment: who/what will be around longer: Trump and republican-controlled governance ... or liberal-controlled academy? Sure, an administration may leave lasting, maybe even catastrophic legacy, but is anyone under any illusion that they are here to stay? Conservatives sure laugh at any such delusions. Heck, they're still reeling from the surprise that any of it happened at all. [They're still talking about Hillary because they don't know how to talk about anything else.] They see nothing but writing on the wall, and it drives them to scary places.
I write all this as a confessed liberal [as if that even needed saying]. But I try to be one with conservative proclivities -- all the best (and largely abandoned) values if I can help it. They do need to be preserved. And we're already in trouble when such a thing must be raised as a concern.