But the point is that they have no legal right to enter them.
That depends, in almost all cases, on whether the service provider/premises owner makes the service single sex or choose your own gender.
The Genderists are winning because they are convincing organisations that it is mean/damaging/unlawful to offer single sex services.
I'll bet that what Sturgeon said is true of most women's shelters in Scotland- because funding gets withdrawn if they try to be single sex.
It's true of Girl Guides because their legal advice is that it's too risky to hold the line.
It's true of several shop changing rooms, who don't want the negative publicity the Genderists were so good at producing.
It's true in prisons FFS.
All this Genderists have achieved despite the single sex exemptions to the EA still being in place, despite a duty on state organisations to consider the impact of policy on women.
What are the additional trans rights that will reduce transphobia?
The only use I can see is to get around the places that are trying to provide single sex services. There will be no situation where the EA can be applied to women, as a class, once self ID comes in.