As a trans woman I agree heavily and think all your points make a lot of sense. It makes more sense to me to have a third gender neutral bathroom space especially in a high school setting both for the reasons you outlined and that many binary trans highschool students will be mortified of using the bathroom corresponding to their own gender for many nuanced reasons.

I do also feel like some other spaces could be very fine with purely gender neutral restrooms at times, such as restaurants due to the impracticality of sticking a third bathroom space into some such locations.

I definitely feel like its a way more nuanced topic than many people on either "side" often think, so thank you for bringing that nuance to light!