Which is true, but then there's the experience I had at my summer job, and I have no clue what that experience proves.
Here's the scene: our office is now twice the size it used to be. We moved into the space adjoining our old one and knocked some large doorways in the wall between them. With twice the space and twice the people working there, it made sense to have twice the bathrooms. Since the plumbing was all in one place, the contractors added two to the existing two, back-to-back.
Thus, from each half of the office, there are two readily-accessible bathrooms. Each is its own full room, with a door that locks. Each contains a single, standard, unisex toilet. So, from an a priori point of view, each bathroom could be whatever we decided it ought to be, gender-wise. So, you might expect one male and one female bathroom in each half of the office. Or, in the casual atmosphere of San Francisco, you might expect a catch-as-catch-can first-come-first-served system: everyone uses whatever bathroom is free when they need to use one.
But you, if you're anything like me, probably wouldn't expect the office to have one pair of neutral bathrooms and one pair of gendered bathrooms. But that's exactly what it had. On the side where the table where I worked was, we all went for the closer bathroom if it was free, and the further-away one if not. But on the other side of the office, the bathroom on the left was girly and the bathroom on the right was boyish.
On my first day, one of the attorneys mentioned these customs to me. She was completely offhand about it: this was how people seemed to treat the bathrooms. She had no idea why; there weren't any rules about it. People even made jokes about fluid gender identities on signs on the doors (one door, on the boyish room, had a picture of "a man in a kilt" AND of a "woman in pants" on it). She didn't mention it again. No one did. No one cared.
And yet I never once turned left when choosing a bathroom on the far side of the office. At the same time, many days, I'd wait outside the bathroom on the near side of the office, and then go in immediately after one of the women in the office left.
My sense of things is that people are (or would be) much less comfortable with people in the "wrong" bathroom than they are with bathrooms that simply mix the genders from the get-go. This state of affairs makes perfect sense: a man in the women's room presumptively has some kind of improper motive for being there. But if the bathroom is marked as mixed, then "ordinary" men go in there, and it's no longer necessary to abduce a "deviant" motive for a man's presence.
Draw your own conclusions.