Trump is a misogynist, so his support on this is for completely different reasons than the views most here hold (and in any case Trump doesn't hold values other than self-worship, as far as I can see).
It's a bit like him wanting the meat he eats as veal to actually be veal rather than pork identifying as veal. Many American right-wing men hold equally dehumanising ideas about female people as the inclusive lot which calls as by our body parts.
So no, Trump doesn't care about women's rights here. He just wants to win again.
In the US, the extremely polarised political situation does mean that it becomes much harder to convince those not on the right in politics that women's traditional rights and concerns do matter without looking like someone who supports Trump etc.
The problem is that now both political extremes view being a woman as a bundle of sexist stereotypes, but progressives can't accept that this is happening to them, too. So they will hold on to their righteous crown as justice fighters for the most marginalised minority ever, while also accepting the idea that women's names, words, spaces, awards etc. can all be erased. They justify this as fair because they are still for the reproductive rights for all people, not only for men, but also for vulva people and womb-carriers and gestation-capable people etc.
Sigh. The situation in the US is ghastly.