Europe solved this problem ages ago with pay toilets and regular police patrol. Some even clean themselves which has an added benefit of dissuading long term residents. The problem has been solved but city's avoid this because NIMBY's will say it incentives drugs, sex, homelessness, etc.
Citizens do not trust that the government will maintain and police them properly, and I think the homeless and sanitary situation in San Francisco is a perfect example.
The toilet situation in Europe is pretty bad from the few places I've been: France, Italy, Greece, Spain, UK.
You can't find clean, free toilets and there is a line to use many of the toilets all the time. Most toilets do not have disposable seat covers, some don't even have toilet seats, some don't have toilet paper, etc. It is a sad state of affairs and makes me miss the much better (though still lacking) situation in the US.
Perhaps it is area dependent in UK. When I lived in the North I think access to toilets was a lot worse than down South. We went visiting to many tourist places like the east coast, Yorkshire moors, various historic places. All these trips had to be planned carefully as usually there were no toilets available. Even in food places in such tourist areas. The only place one could always find a toilet at were petrol stations. Another thing is toilet availability for people going for a night out.
In contrast when I lived in London I always felt access to public toilets was pretty good. There were lots of places like McDonald's, Starbucks etc that always let the public use their toilets for free. Then there are free toilets on many large rail stations past the barriers as well as paid toilets before the barriers. Finally there are free standing paid toilets dotted around the city. Back then one had to put a coin to open them.
Personally I think London is one of the better European cities in this regard.
Citizens do not trust that the government will maintain and police them properly
This is the problem not NIMBYs, It is the correct assessment that the bathrooms will not be policed or maintained. If they were policed and maintained, no one would even notice them.
The problem is that some people do not want pay toilets. They want free public toilets and would call Europe backward/racist/whatever for making people pay for toilets.
I won't assume what people would call "Europe" which is a big place. However, it's certainly true that pay toilets are not a thing in the US in general and certainly not in train stations, restaurants, etc. as you at least sometimes see in other countries.
It's probably complicated at this point in that I assume most people in the US don't carry change today and tap to pay small amounts is fairly nascent.
Even if you want to pay, it is hard to find toilets that are clean and do not have a big line. It is a horrible situation, especially when visiting with young children.
If a person were begging outside of a toilet I'm sure someone would help them out until it becomes a regular hustle like people outside of grocery stores is.
Because they don't care, can't be bothered to wait for one, and any ones with personnel are probably not going to admit them because they will make a big mess.
Someone who needs to pee is going to pee, whether they have a dollar to spare or not; I'd rather they are able to pee in a restroom rather than on the grass, sidewalk, or someone's doorway. It is cheaper and healthier for society as a whole if people do their business in restrooms, which is basically the reason why we've converged on a society where toilets exist in the first place, rather than fleets of trucks circling around and washing the streets constantly.
Public toilets in France are frankly horribly maintained and smelly (wtf is wrong with ventilation there?) that San Francisco is a toilet paradise in comparaison.
Citizens do not trust that the government will maintain and police them properly, and I think the homeless and sanitary situation in San Francisco is a perfect example.