exactly - it is a bathroom, what is that big of deal?

these gender divisions are arbitrary - Hasidic Jews have sexed religiously services (women separated from men) and Reform and Conservative Jews have co-ed - both kind of people seem to survive it, which proves that both arrangements are feasible. Muslims have sexed services too, but Christians do not. Clearly, both arrangements are feasible.

There are co-ed schools and girls' / boys' schools - likewise, it proves that both sexed and non-sexed (co-ed) arrangements are feasible.

I think houses of rich people in Ancient Greece were separated into HIS and HER areas, but most modern houses do not have this kind of a separation - lucky people have enough room for a HIS closet and a HER closet, and the less rich couples just somehow divide a closet betwixt themselves. And still survive!

There are REAL reasons for sexed arrangements in sports, though - you cannot have women basketball players compete with men, for the most part. But bathroom use???