On the 'what women/men do in toilets' question:
There's also the fact that toilets by their nature are enclosed areas shut away from the main part of a building, and sometimes there are only one or two people in there.
Being shut in a room with a strange man is a potentially risky situation for a woman. Currently, or at least up until gender ideology took over, seeing a man there was a red flag. You knew that chances were he was up to no good, and could react accordingly. You would get out if you could; or lock yourself in the cubicle; maybe hang around to see another girl out safely; call staff and ask them to remove him.
In the proposed approach where anyone can go in, we're supposed to just ignore a man there. We're not supposed to take action to keep ourselves safe. We certainly wouldn't get support from staff, we might even get called bigots and thrown out of the building ourselves.