It still doesn't matter if you don't always know. Not always knowing does not mean that single sex spaces are unworkable.
We know this because our entire society works mostly on taking people at face value, and questioning people who don't meet the criteria asked for.
I was constantly "misgendered" when I was a child based on short hair, unisex clothes and not having grown boobs and hips yet.
If I'd wanted to go to Cubs (back then it was single sex) I might possibly, if I'd tried really hard and wanted to lie about it, been able to go, certainly for a short while. If I'd been found out then I'd have been asked to leave.
So yes, a single-sex WI might potentially have a transwoman join and the deception not be discovered. You're right, it's more likely if it's your Mermaids woman's child who'd been on puberty blockers than if it's Pat who started expressing their true self in their 50s. It might happen.
But just because Jackie joins and isn't found out, doesn't mean that we lose the right to say to Pat, you are a transwoman and this is a female only space. I'm sorry that this hurts your feelings but that's the rules. And if Pat happens to be a very masculine looking female then that is easily cleared up.
It gives the Pats of this world terrible sads that the Jackies sometimes pass. They used to call them "Truscum" (might be passe now) and demand that we treat them all the same. Which I'm happy to do - no access for any of them, and if they want to lie and cheat well that's on their conscience. We don't need a 100% success rate to say no.
The thing is though, they don't want to lie and cheat - this is all about getting us to throw open the doors and welcome them all in as women exactly the same as us, and I feel like your argument on this thread "you can't always tell" is trying to do the same thing.