re: Do/did the Japanese have unisex public bathrooms anyway?
No.. i didn't see any. most of the baths aren't unisex anymore either. (but i did go to one that was unisex, but my sister and i left as men started to arrive.) at least not in cities.

though nudity in general is not a issue, it is strictly taboo to expose pubic hair, (even women in a womans bath would make sure to have her hands and a wash cloth covering her self in a public bath)

Tokyo has lots of clean public toilets availlable, but none provide tissue. everyone is expected to carry their own. some of the bathrooms have western style facilities, some are old fashioned "holes in the ground" with small platforms for you feet, and a small handhold to on the right side to hold one while you straddle the hole.

it rarely get below freezing, so they toilets are open year round. (they only have cold water for washing your hands-- and no towels or hot air driers..)

i only traveled in cities, so i don't know what things are like in small towns or the country.