Would the sinks still be segregated? or installed inside toilet cubicles?

I have seen parks where toilets were in individual little rooms and sinks were outside and public. I have also seen arrangements where each little room had both a sink and toilet. They both work okay, though it is nice to have the sink be private as well. I don't see any reason why we can't work on coming up with new standard patterns to try to address issues like cost and space efficiency while improving privacy.

As for the existing toilet situation and why it is the way it is, my suspicion is that it was something that worked reasonably well historically when more than half of all people were rural residents. Sometime in the lifetime of my children, we crossed the point where more than half of all people live in cities.

Things that work reasonably well in a relatively homogenous small community or rural area don't work so well for highly diverse big cities. Times have changed. The ongoing national debate about bathrooms makes it clear that our current setup is now problematic.

I fail to grasp why you are so hung up on this detail. Historic precedent is very often a matter of "it was what they could afford in a poorer society than modern America" or "it was the first thing someone thought up without really studying the problem space in earnest." The fact that this is the status quo in no way whatsoever suggests that it is some kind of best practice that we should dread altering.

When I was growing up, I lived on the edge of town. In one direction, there was more housing, plus shops and businesses. In the other, there were wooded areas. It was really common for men to relieve their bladder on a tree somewhere while out and about. I also learned to relieve myself in the woods, though it is less convenient for a girl.

In a situation like that where it is logistically feasible and socially acceptable to just pee on a tree, there is less need for public toilets. But that does not translate well to the big city. You don't have sufficient privacy nor sufficient greenery. Peeing on a tree occasionally in a wooded area is not a threat to public health. The tree will consider it to be fertilizer and it will not cause the spread of disease amongst people.

But in the big city, if lots of people pee and poop on sidewalks or in alleyways, this is a public health hazard. There is currently a hepatitis A epidemic in Southern California that is killing people. It is related in part to large numbers of homeless people relieving themselves exactly that way.

Public toilets are just less of an issue in rural settings. People routinely just pee on a tree. In some parts of the world, population densities reach small city levels, but they are too poor to build the kind of infrastructure for modern bathrooms. People continue to relieve themselves in a nearby field. One result: Women get raped and sometimes murdered while trying to pee and poop.

Bathroom facilities are a real need in highly populated areas in a way that they just are not with fewer people. We now have 7 billion people on the planet. This is unprecedented. It should be no surprise at all that it leads to unprecedented problems in areas that no one would have ever expected.