A few thoughts on this...
1. I'm not sure it's going to be practical for say a gym owner to introduce a birth certificate check for every registered member - so in effect the default has to be self-id/honour system anyway.
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.
2. Not sure how far back we are going in 'having a right to eject people' but you are going to have to tread very carefully around anti-discrimination laws to eject people or even ID check people as things stand currently.
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.
3. The laws against bad behaviour may be difficult to enforce in some respects but this is a general feature of laws against bad behaviour. If we are talking about a private gym though I would imagine that the owners have significant leeway to exclude people who appear to be creating issues provided they can show a justification if called on it. Like everything else though they leave themselves open to a lawsuit if they are careless.
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.