Is an organisation which does NOT provide single-sex spaces which are needed by women who have PTSD (a disablity) or women of certain religions, or just women generally, in breach of its equality duty?
It seems straightforward to me, once you strip away all the bullshit. If my local pool is allowing penis people into the women's changing rooms I don't want to go there. That's because penis people are dangerous to women and it's uncomfortable and undignified to be expected to strip off my clothes in front of a penis person.
If the pool can't provide safe spaces for women to get changed, then they are making their facilities inaccessible to women. It should be quite easy to provide evidence of this; there are loads of examples of women ceasing to use gyms, pools, yoga studios etc. because of penis people in the female changing rooms.
And that's without the additional protected characteristics of religion or disability where it's even more clear that not providing female-only spaces is preventing people with protected characteristics from accessing the facilities.
It would have been useful to have guidance on this from the EHRC.