The ancient romans actually didn't have gendered bathrooms. Or separate stalls. Just everything hanging out together, watching each other piss and shit. And then they'd all wipe their butts with a common sponge on a stick.
Gender is a scam sold by lazy thinking patterns just like stereotypes.
It persists so readily because most humans are reactionary buffoons who would rather the world fit their expectations than gleefully explore things they don't understand.
Yes, most humans are pathetic losers who give in to the sweet lazy comfort of not thinking through every detail, because it is painfully obvious that gender is not binary, nor even sex.