For 95+% of people, a simple division by sex in facilities like restrooms and locker/shower rooms works. And there is an option available for everyone, even if it's not their preference.
This kind of pontificating makes it hard to take you seriously. In a 50K seat football stadium, dividing the restrooms by sex doesn't make them "private". It's just being reasonable. You cannot just redefine words to suit your personal issues and expect the rest of us to give you special rights.When it comes to the public's accomodations, the correct answer is (spaces for any single one), (spaces for nobody), (spaces for anyone who wishes) are acceptable. (Spaces for only some) is not 'public', at all; that is 'private'.
Tom
Unfortunately, whichever facility you send the trans people to there are issues.
Thus, how about a different approach? Build three facilities. Male/Female/Any. Any means any--it's not restricted to trans people. I think we would soon see that most people would regard it as a non-issue.
If you're building big facilities it's not going to make much difference--it means another door but little other costs as you still need the same amount of space and equipment. If you're building small stuff you've got what we see now--family restrooms.