I would chalk it up to a vestigial form of white patriarchy showing up when the concept of the public toilet got invented. Federici points out in Caliban that white womanhood in particular and womanhood as a patriarchal subclass in general are forced to both present and materially be weaker than the men around them. Women weren't allowed to wear pants because the relative vulnerability imposed by their culturally acceptable mode of dress was part of the point. Women who did gain independence were burned at the stake of course. So if the construction of vulnerability, both socially and materially enforced, put women at a disadvantage, than it stands to reason from the reactionary viewpoint that therefore women need chauvinism, they need protecting. So if you're designing public restrooms, you're modeling it on the norms of the European nobility most likely, you're aware at some level both that women are made to be weak and men are given instruction on how to advantage that in a society that accepts most violence against women, then you get the age old turn of needing to now protect the women you've weakened. You give them their own bathroom to protect them from all those nasty violent men out there that patriarchal societies spend so much time building.

I do use the urinal with my genitals hanging out, to be fair. I could see ungendered bathrooms if they replace the urinals with stalls. OR I could just get used to it but there would be some getting used to it.