There are several reasons third spaces have not been campaigned for.
They are being campaigned for. Transsexual journalists/bloggers Miranda Yardley and Debbie Hayton head the campaign.
Firstly the custom and the law developed in which it was quite usual for trans people to use spaces inline with their gender and no-one seemed to care until a couple of years ago.
Again with the refusal to accept female no. There is a reason why female people have been pushed to put down boundaries, it wasn't something they randomly came up with one day.
Secondly third spaces would force people to out themselves and could increase bullying and harassment of all gender nonconforming people - there's a space for you freaks now get out of here etc.
Your choice of language implies what is often the real issue for third spaces, that TW do not want to be 'othered' from female people. However male bodied people are other from female people and it should not be the case that female rights are overturned because male people do not like those boundaries to their freedoms. Female people have rights too.
Gender neutral spaces would obviously be used by TW, TM, NB people and plenty of people such as those women such as Nicola Sturgeon who spend a great deal of time scolding females who want privacy and are delighted to be undressed in mixed sex spaces.
This is not a reason to force female people to suck up the challenges and discomfort of losing single sex spaces. Again: third space offers everyone two choices. Forcible commandeering of female only spaces excludes some females from any option. That's not an acceptable solution.
Thirdly, shared spaces, which could still be used by male born male identified men, would do little to reduce trans people's safety concerns.
Hang on you keep telling female people that there's no danger to male people in female spaces and nothing bad ever happens in mixed sex spaces. You can't have it both ways. Those gender neutral spaces have yet to be designed: design safe ones. Single rooms with basins. I'd suggest doing this for female people who need female only provisions and let male people turn the women's mixed sex and the females have a female only option for women who need it.... but we all know male people will never accept female people having boundaries or a space that they cannot access. Which is now why Hampstead Heath offers male people a choice of three ponds, women who can access mixed spaces a choice from two, and women who cannot access mixed sex spaces no pool at all.
Not exactly 'inclusive' or 'kind' or 'intersectional' really. In fact deeply sexist.
And fourth, it's an unwinnable campaign. It would cost hundreds of billions. It might even be impossible in some older listed buildings.
Nonsense. I was involved in the disability act (as it was) implementation. It's easy, the blueprint is there, the money will be there, as it was for accessible toilets. This is after all, the only inclusive solution. And we're talking about charities that have been able to try and change laws in the matter of a couple of years they have so much money and power; this would be a breeze. IF it was what those male led, male centric charities wanted.
Unless your main aim is that female people shut up, lose their rights and let male people do what they like.