Gender neutral obviously will be used by anyone who chooses, as discussed here many times - people with opposite sex kids, carers with opposite sex clients, all the women who believe in validating men by exposing their bodies, so the 'outing' bit is rubbish. As usual, the consenting women's presence and bodies won't do.They're not the ones desired.
Leaving aside the many issues with commandeering disabled toilets, they aren't 'outing' either unless being disabled is a shameful thing? Many use them who aren't wheelchair users for various reasons.
It's the usual thing of a desired solution frantically flailing around trying to find a convincing causative reason. And the issue really is that if women are permitted a space that men who identify women are not allowed to go - even if they have lots of accessible provision themselves - it means that everyone is saying those men are not 'real' women. And women are being allowed to escape and have things those men can't commandeer and own. Like Sarah in Brighton with the rape crisis service - for a single sex option to even exist was too offensive to men to permit alongside their three available options.
The boundary breaking and being not just the very clearly always most important but the ONLY group that matters is a major part of this culture. And with this absolute inability to even acknowledge the existence of women's needs and rights, never mind considering ways for win-win solutions, they don't have a hope.
The bottom line is the belief that women wanting single sex spaces is a wholly irrational, unreasonable demand. Men's feelings about this are a terribly sensitive issue - Labour are running around like headless chickens about them - women's feelings and actual exclusions and assaults, YEARS of them, pfft. They're only women. They're not really human or anything.