winjim
Welcome yourself into the new modern crisis
It's controversial (to some people) because it shows you acknowledge that a man who identifies as a woman isn't actually a woman at all - because their 'woman-ness' depends on the situation. That's considered transphobic by many trans activists.
I've never heard a feminist say dads shouldn't take their daughters out. Most women would be ok with a father taking a girl into the women's toilets if needed, though there's often a baby changing toilet these days. The toilet situation is easily solved by a 3rd unisex space of fully enclosed cubicles each with a sink. And additional separate toilets just for women. Unacceptable to the trans community though.
You keep speaking for 'most women'. I'm not sure that's wise.
The twitter thread I'm referring to has fathers waiting outside the ladies' or simply not taking daughters out at all, I can't see any mention of fathers going into the ladies' toilet. Are you sure that women would be happy with that? It would make me uncomfortable, I would definitely feel I was intruding. My preference would be to use an accessible / family room if there is one but if not I would take her into the gents'.
University of Manchester has some good non-gendered toilets. Big wide entrance onto the corridor so it's visible, no nooks and crannies, busy part of campus. Cubicles with full floor to ceiling doors, shared sinks in the middle but TBH I think a sink in each cubicle might be a bit impractical, would take up a lot of space. It seemed well designed to me.