The problem with both sides of the locker room issue is not that one side's solution is unacceptable. It's thoat both side's solutions are unacceptable.
Even worse, instead of trying to address the shortcomings of one or the other and find something workable that is as fair as possible, each side takes the easy way out and just continues to point out the problem with the other side's proposal.
The idea of checking ID on everyone who uses a locker room is ridiculous. Birth certificates are absurdly ridiculous.
At the same time having no criteria or means of enforcing a criteria when one is suspected of violating it is also unworkable. Unless you want to just cut to the chase and go to unisex facilities. (That's what these spaces become enforceable as.) The (admittedly small) number of people who wish to violate social norms are not going to be deterred by social pressure. And that's the case currently. However, a facility manager can ban or trespass someone who is violating the norm. And that is a deterrent even if its a wrist slap.
There's an imperfect medium available. One idea is that there are criteria, but it's not routinely checked. If someone thinks that criteria is not being followed, they can inform management, who can make their own assessment as to whether to inquire discreetly. As I said, it's imperfect. But I haven't heard of a better proposal. And it's essentially what has been in play in the past.
Or we can just leave things the way they are. No bathroom laws, but managers can enforce policy.