My thoughts on this aren't very well sorted out and I am hoping that some wiser women can help.
I think there is something qualitatively different about the rights demanded by TIMs and TIFs, but TIMs try to conflate the two.
When TIFs insist that "men can get pregnant" and ask us to say "chestfeeding," it is obnoxious and denies basic reality. But it does not remove women's safety the way that demanding access to women's spaces does.
So I think what is happening is:
TIFs convince society that pretending men can get pregnant is the kind thing to do -- we all know that men can't actually get pregnant, but this makes TIFs feel better and we all want to be nice. So we agree to be "inclusive."
TIMs capitalize on this and demand access to the same spaces because we're being inclusive, aren't we? It would be MEAN to exclude them and we already agreed to be nice!