However, in the US almost everyone says bathroom, even at work.

It starts at an early age by asking children if they need to go to the bathroom. "Going to the toilette" seems very crude to many Americans, and using bathroom as a euphemism for the other thing that happens in there seems more polite. Yes everyone knows what you are doing, but we can pretend you are just "washing up" This makes very good sense at a restaurant, for example. You are going to wash/bathe your hands. We won't think about what else you might be doing.

And, technically, as long as there is a sink, it is, historically a bathroom. For much of western history, bathing meant a bowl of water and a cloth. Fully body immersion was often believed to be unhealthy in many regions and times. So a bathing room, was by and large a place with a wash basin. Obviously certain cultures had communal bathes, which did not involve a toilet nearby.

Now, in fact, this all plays very well into the authors article. Fear of disease transmission in early factory toilets led for calls to modernize toilets and "protect women" "the weaker sex" from the dirty, disease ridden men who worked around them.
Total macho bullshit.

I do think the urinal plays a big part in this, as you say. But if you look at old bars and sports facilities, places women often didn't go; you find the trough urinal. For high volume, it is extremely fast and space efficient. And interestingly, the trough urinal concept goes back to Roman times if not before.

But yes, that being said, it has certainly been used in the last century or two to marginalize women. In particular, potty parity is particularly egregious.

I think that the common bathroom is a great solution; they are starting to come along, at higher end locations often still segregated, you have a common sink area, and private "water closets" literally.

Now this is great for potty parity if you desegregate; except for one thing and the reason I still advocate for urinals.

Men, by and large are lazy and sloppy. Women like to complain they never put the seat down...well actually what's worse in public toilets is when men don't put the toilet lid up before urinating and leave a wet trail of urine on the toilet seat.

So, I would argue, there is value to having urinals, if only to keep seats clean. Not sure how to resolve that other then to tell guys to get the freak over their paranoia and insecurity.