I don't have a problem with that anyway. But there are still plenty of those bathrooms out there, and no one I've ever heard of was so offended that they ran out and filed a lawsuit that I know of.
All of this is manufacturing division for political gain. White/Cis/Straight identity politics. Unfortunately, appealing to the worst in people pays off at the polls a lot of the time, especially when the target is so vulnerable.
Here, the law is supposed to work this way: "Your rights end at my nose." This means that if there is no harm, there is no imposition. Since our system is the child of the British one, I'd be surprised if that basic principle was effectively different. This stuff should be Unconstitutional so long as there isn't any demonstrable proof that Traswomen in bathrooms were fundamentally harming the rights of someone else.
But that's not what we get, of course. The people pushing this are claiming "harm" being feeling icky and offended. But there is no right to not feel icky and offended. You would NOT say anymore that segregated bathrooms (a direct comparison with the same "icky" justification) were Constitutional because of all those Black women who were offending by their presence.
They claim Trans women should not compete in women's sports because they will have an unfair advantage. It's funny when women push that because the underlying justification is that it's unjust because women are just weaker and less capable than men, and that's why Trans women should be banned. "...it's not fair..."
If we were really after "fair," why are women barred from the prestige, opportunity, and most definitely the money in men's sports? Is it not fair to let everyone play, and if they can't cut it, they don't get paid? I know for a fact that the championship women's soccer team here would wipe the floor with the vast majority of men in that sport. Why should they get less of everything? THAT is what's unfair.