If the genderists weren't demanding access based on gender identity, why would the Republicans have even needed to pass any such laws?
That's one of the first things that puzzled me. Why, in the 2000s, would Republican states suddenly be doing this? Hadn't women and transwomen muddled along happily up till now? Was there a problem? A bit of a niche random thing to start a political fight on, surely?
Having seen genderists, I rapidly concluded - "oh, I see". They saw what the genderists were doing, and decided to push back. Maybe before there were many real issues on the ground, but it wasn't out of nowhere.
Toilets still is a bit of a weird battleground, but I guess the point is that they were historically unregulated - sex separation by convention rather than law, so start enforcing it. Other things (prisons, sports) were still sex-segregated by law/regulation, so nothing to fight there at the time.
At least until Obama did "for sex substitute gender" at the end of his term, which Trump immediately reversed.