It's interesting: from your perspective, the concern is people clinging to the phrase "objective truth" to mean "tradition" and that anything that threatens tradition is a fundamental threat to reality. From my perspective, the concern is people rejecting the existence of an "objective truth" as something that oppresses their wants and needs.

I'll bet your father believes in the greenhouse effect. He probably even believes in greenhouse gasses. Show him a picture of Antarctica with a pink blob over it and he'll know immediately he's looking at a hole in the ozone layer. The pieces are there, it's the ideology that ties it all together that he rejects. It's hard. My earliest memories of educational television include Mr. Wizard warning us of the impending new ice age. Weekly Reader was all about how acid rain was going to destroy all structures by 1990 and we'd all need to wear SPF1000 sunscreen outside of our silvered biodomes because gamma rays would have sunburned all living matter to death but it would only matter a little because by the year 2100 Venus would be a more hospitable climate than Van Nuys. The population explosion would lead to widespread famine by the early '90s and there would be no petroleum available by the year 2000 because we would hit peak oil by 1978. This, of course, is assuming that we wouldn't die in a nuclear holocaust brought on by the Contras.

So I can see how someone could retreat into a belief that climate change is fake. This is probably the reason so many discussions of climate change have been so pedantic for so long: any consumer of media has seen so many nightmare scenarios that they don't bother sleeping anymore. And when we discuss "climate change" we're talking about a constellation of observations that lead to a damning conclusion. If you accept that damning conclusion the constellation makes sense. If you don't they're isolated incidents that you explain away as irrelevant.

"Sex" and "gender", on the other hand, have not been widely acknowledged as individual concepts for long at all, at least not in the mainstream. Not only that, straight white males are invariably wrong in these discussions and generally subjected to ridicule. Someone expressing an "objective truth" about cisgender women is someone refusing to use the word "cisgender" because they've never had to before and no one has ever made them feel bad about it.

In both cases, the party in question is saying "I'm not going to argue about this" (because it hurts my head). They're asserting that the argument is invalid because the subject is beyond argumentation. The problem is it's being conflated from the philosophical (what is girl if not female) to the practical (I paid you back the $20 I borrowed because I feel like I don't owe it to you anymore).