Off the cuff generalizations require either (a) the assent of your interlocutor or (b) convincing data showing that the generalization isn't hasty. You have neither, here.
"Off the cuff generalisations" LOL
So, firstly, are you actually trying to claim that more gyms/pools/sports facilities than not (across (say) the USA, or Europe, or Australasia) currently have unisex changing areas? Is this actually your position?
Is that position influenced in any way by the fact that you happen to currently use a facility which has a unisex changing area?
What proportion of sports/gym operations that you've ever used in your life had a) unisex changing facilities, or b) separate men's/women's/disabled facilities?
You quote one single unisex facility (and I myself have enountered one such facility myself, as I said), and then somehow seek to use that to claim that it's an "off the cuff generalisation" to state that the clear majority of gyms/pools/sports centres etc have facilities which are segregated by sex??
I suspect that most people in this real world we all inhabit know full well that the vast majority of gyms/pools and sports facilities (whether in the USA or anywhere in Europe or Australasia) currently have changing areas which are segregated by gender and disability.
But if you want to carry on arguing that point.... be my guest.
"The assent of your intolocutor" LOL
Either "Family Changing Room" for the large ones with certain amenities or simply "Changing Room" otherwise. No need to call out sexes, genders, or range of physical abilities if we're not limiting who can use the space.
So you'd have
"Men's changing room" on one door.
"Women's changing room" on the second door.
"Family changing room" or "Changing room" on the third door?
(And remember, we are specifically discussing how you'd change the label on what is currently labelled "Disabled changing room" in a facility with men's, women's and disabled changing rooms...)