That’s a bit of straw. What do birth certificates have to do with anything?
I’m not sure about where you live, but my Illinois drivers license is my legal ID. People who don’t drive get a state ID card, which is identical to the drivers license. I have to present it to buy alcohol or tobacco, write a check, board an airplane, and several other things.
In fact, I had to present it to sign up for the gym. (Must be over 18 to sign the contract and also to use the tanning beds.
The idea of carrying birth certificates is silly. No one is proposing that.
Legally, you are allowed to eject someone, but you need to be able to show a reason. You can confront someone, but they aren’t required to show ID. (Of course, the gym already has my ID information to support their reasoning.)
What you are saying is that the segregation of locker rooms is already unenforceable. Which means a cis man, effectively can use the women’s locker room if he likes. Because the only thing stopping him is social pressure.
So all facilities are effectively unisex.
It seems like most people on both sides want segregated locker rooms and expressly want to keep cis men out. But any tool that would accomplish that is objectionable.
And that’s the problem. Without a clear objective method of determining who has access, there is no way to enforce it. The ID cards most everyone carries has your legal sex/gender on them, so on its face it’s an easy way to resolve questions. Some women (both cis and trans) would be offended if asked to show it though. So I don’t see a solution.
Lack of a solution does not make the problem disappear, however.