That is just selection bias, people who aren't rich have to work and therefore you meet them even when they don't want to socialize. Rich people who don't want to socialize don't need to, so you only see those who really wants to be out there and meet people.

Well, in my case, when I bumped arms with the rich and well off was mostly back before I was 25. There's some occasional intermixing between the talented peasants and the kids of the wealthy at things like prestigious scholarship trips and the like.

This would probably hold true if I were going just based on the rich people I knew through work since most of them I've interacted with as a representative of my organization (which they have some interest in), so you'd be correct that there's a motive there.

You have touched on one of the biggest reasons I am against private school. We can't force our countrymen to know each other as adults but I think they should start in the same classrooms.

I go back and forth on the question. I definitely think we need to get rid of the two-tier system that's functionally 'pay to play', but I don't think just public schooling is going to solve the issue, because the problem with putting either the peasant Einstein or oligarch's kid in a normal school is that their arrogance is almost never curbed, because there are a lack of peers to call them out. The exception to this would be areas where those groups cluster together: A Silicon Valley or DC public school might have enough privileged kids that they can call each other out.

On the other hand, I'm a talented peasant who went to a public school system of ~30 kids per grade. Setting aside IQ's usefulness as a measure, I had an IQ of ~145, which meant I was not socializing with the other kids the way I should have, and it also meant that (once the tests identified it), most adults were not interested in addressing my arrogance or mistreatment of other people. I very clearly remember in late elementary school, I viewed my classmates very similarly as I did to intelligent dogs. I can see a very similar thing happening to the one wealthy kid in, say, an Appalachian school. Sure, you're all nominally equals, but if the teachers need to go to your parents for help with the school supplies, you're not going to be disciplined in the same manner as other children, and wealthy parents teach their children to use their social status.

I'd be wary of solving social isolation between classes by opening up more ability for local fiefdoms to pop up. (Think of local business owners that are POSes to their employees because they know their only other option is Walmart.) From the point of view of the average citizen, neither type of ruler/elite is good to deal with.