^Yes, on the whole I think you are right (although I think you mean Transwomen?). A lot of this is about power, really.
Also, how many places have the space or plumbing to be able to comply with that? New builds could do it, but converting older buildings often would not be possible. So instead, rather than have men inconvenienced, the Ladies 'conveniences' double up as 'gender neutral' and often are also available to the disabled and parents with babies, so men, women, transpeople, the 'non-binary' the disabled and parents are all expected to use the space with cubicles only, whereas the men get a place all of their own with both cubicles and urinals. The best solution? You Decide.^
Sorry DD, I had a moment and find quite a lot of the language around this quite confusing!
I'm not sure that I have the brain power to decide but agree that women draw the short straw where public toilets are concerned.
As for "asking" anybody what gender they are, quite often you don't have to ask, it is very obvious. However, if somebody was behaving themselves, I'd have no reason to ask them and would live and let live.