I wanted to pick up a further point from your 'men in positions of trust and authority' argument. It would seem to follow from this that you are saying that, since a proportion of these men will use their positions to harm women, we might as well even things up by allowing women to be harmed everywhere.
Surely the sensible reponse is to say 'let's improve safeguards, oversight - and most of all believe and take seriously people when they complain about a man's behaviour - so that harm by trusted professionals is curtailed'.
You seem to be suggesting that the people unhappy about transwomen accessing women's spaces are not making that case. I don't see that lack. Post-Saville, #metoo (and Shipman) etc, surely everyone is aware and supportive of the need to tighten and enforce safeguards, everywhere?
The only people moving in an anti-safeguarding, libertarian direction are transactivists.
You make a very similar argument when you say that men who want to harm women will find a way in to women's spaces to do that anyway. Well, no, they won't, generally don't and pps have explained repeatedly why that is. Because people, women and men, will uninhibitedly challenge men trying to enter women's spaces. Everyone seeing such an attempt is immediately on alert, noting that 'something's wrong here, what's this bloke up to'. I've seen this happen, with a pissed bloke going into the women's loos met with an immediate, stern 'Hey, this is the ladies!'.
You might indeed just as well say 'well determined paedophiles will get to children anyway, so why bother with gates on schools, or regulating who may enter? Safegaurds and checks are in place because they work. Cultural reinforcement, noticing, challenging or reporting individuals who look like they're in the wrong place, plays a huge part in this.
At the moment, with so few transsexuals, who all make a big effort to look like women, everyone can see that the person, while natally male, is at least a determined cross-dresser, not any old man. Then there's the 'trying to fit in' behaviour.
Of course TRAs want self-ID, because they want rights. The right to enter, the right not to be challenged - and the right to disparage anybody who dares try. They don't want to go to all the effort of trying to fit in, visually, behaviourally, or socially - being aware of others' comfort and preferences and acting to accomodate these, like most women do.
So I'd suggest that answers your 'what's the point of a GRC if they're never shown' question. Those who have them wouldn't generally need to be challenged, because their behaviour would be unlikely to invite challenge.
That doens't mean that some women won't feel uncomfortable that they're present.