The "coddling" one has always been a bad faith argument; a post hoc rationalization for political goals.
For 11 years now Haidt has very deliberately identified as a centrist, partially because folks like yourself unfamiliar with his work find habit in dismissing his research as being politically motivated. Just because a researchers findings don’t align with YOUR politics doesn’t make the research itself politically motivated, it’s more likely your worldview was politically steeped from the beginning.
I don't see how Haidt identifying as a centrist has any bearing on anything; the idea that "centrism" is apolitical is also untenable.
As far as centrism goes, you're totally right, it's a political ideology like all the others. What I was trying to point out though is that despite being a very vocal American social critic, he's staunchly deciding _not_ to take a public position on American politics. Despite this, I've seen people on either side of the aisle still accuse him of shilling for the other side, a take I've found super exhausting. I'm not in total endorsement of all of Haidt's research or methodologies, but I do think he's operating from a place of scientific earnestness and honesty. Framing him as politically motivated is inaccurate.
Centrism focuses on feasibility, and on incrementalism. The possibility that We're Just Doing It Wrong isn't a part of that worldview, and so should it turn out that we are, in fact, doing it wrong, centrism will never address it.
Put differently, centrism is philosophically equivalent to a local minimum/maximum from which any significant change makes things worse than the status quo (and legitimately so). This metric fails to note that there are better minima/maxima that are reachable, but only if things get worse for (hopefully a short) period of time.
Second, the actual role centrist do is to actively work against improvements. They do not say "I think it is infeasible, but good luck". They say "trying to do this is wrong".
Yes, I'm sure such easy accusations exist. Not long ago, climate change was "left-wing", and now that we know that it's just science, people should remember who often scientists that were just doing their job were dismissed for being left-wing.
But on the other hand, on the left and the right, scientists are human and are biased by their understanding of how the world works and should work.
For Haidt, I don't have a specific opinion yet, but there are few elements that make me think that, even if he is not conscious about it himself, he may be politically biased:
1) In a lot of his writing, I never really see him taking bias-control measure seriously. He seems, at the end, despite what he may say sometimes, to consider himself "too smart to be biased", which is not a good thing. Having bias-control measures (aka starting from the hypothesis that maybe my conclusions are biased) is an easy and harmless way to do good science (even CERN is doing it by asking members of the Atlas and CMS collaboration to not exchange on non-published results).
2) The scientific critics are pointing at scientific elements that are legitimate. They are not saying "I don't like these findings, so I will say Haidt is right-wing", they are saying "there is a scientific problem with these findings". The large majority of them don't even say that Haidt is politically motivated. The "politically motivated" is appearing after, when people try to understand why Haidt will prefer some conclusions over others while they both are as scientifically grounded. In particular, I'm far from convinced by his counter-argument. For example, "the map for the territory" argument seems to say "stats is hard, so it's ok to claim we are sure even if stats say we are statistically not sure", or "the pascal's wager" argument seems very political "dropping some left-wing values and reinforcing some right-wing values is just all benefits, because obviously we are better off in a right-wing value society"
3) Scientists that are "neutral" will sometimes find findings that are politically neutral (for example: "I find that butter has an impact on mental health"), sometimes left-wing (for example: "I find that some traditional values have a negative impact on mental health"), sometimes right-wing. But if a neutral scientist is studying different topic and, by chance, always ends up with the findings that corresponds to what a right-wing person would think, this is suspicious. Haidt have worked on different topics, as you say for 11 years, and apparently, his reputation for preferring a conclusion more aligned with moderate right-wing is not due to his last topic, but from the observation that a truly neutral scientist will sometimes align with right-wing, but sometimes with left-wing too.
What the various kinds of "avoid coddling" types want is determining the details of what should be allowed to traumatize them, specifically, in order to shape society.
As far as I understand all Haidt calls for is to remove the boundaries that have been established over the past couple of decades, or what selectivity are you hinting at here that are part of that supposed plan to shape society?
More generally, the discourse tends to boil down to "I was spanked and I turned out fine!".
Yes, there is an inconsistency in his writing regarding why social media specifically is something we should protect children from. But Haidt is not suggesting that a return to a corporal punishment regime, or anything like it, is desirable. His concerns are things like free play, independent movement, negative (not only positive) interactions and so on.