I started to lose the will to live at about page 18 and have only skimmed from about page 10 so sorry if this repeats something.
The real issue, for me, with all these signs is that they are provocative. On purpose? maybe. So if a sign has in nice big friendly letters: Men's toilet ->, Gender neutral toilet -> and then in smaller harder to read letters underneath "there are women't toilets somewhere else") what is the reason? Do they want to "erase women"? Do they want to annoy women? are they trying to trick people into tweeting about women's erasure so they can come back and say (in the words of pp quoting Douglas Adams) "silly! they're down there behind the beware of the leopard" sign.
When i need the toilet, i need the toilet. Now. I'm not a shy retiring violet and on occasion when the alternative is to pee my pants or worse, i have merely barged into the nearest toilet with no queue (usually men's) shouting as i come in that it's an emergency, I'm not looking but i must go now!" and i have had precisely zero complaints about that in the 50 years or so i've been doing it.
So I'm not averse to using the men's loo, and i am also not averse to a GN loo either. But some people are. It is beyond nasty to hide the signage or otherwise obfuscate (for men or women or people who need accessible facilities) where these facilities are.
I use a pub not far from where i live which has a Weatherspoons approach to the ladies - it is down some stairs, then up half a flight of stairs and along a corridor and there are only 2 cubicles. The men's is just behind the bar and has 3 cubicles and 3 urinals. I'm pretty sure that's not actually legal since my understanding is that access to toilets should give parity to the number of facilites that can be used by male sexed and female sexed people. (that could be wishful thinking). Anyway - rather helpfully, above the sign for the men's in equally large letters is a big sign with a diagram of how to get to the women's. Which is more helpful than the signage i saw in the Barbican tweet or that Belfast pub.