I think the greater diversity in public schools is better for a kid's development but that's a US-specific thing
absolutely. why just a US thing though?
Private schools in the U.S. generally are either heavily religious (Catholic schools, parochial schools, etc), or elitist where the tuition is tens of thousands per semester and a family will generally pay more than even tuition at a private university. Not saying that schools and families don't try to gamify public schools into creating their own homogenous and segregated schools, but its just easier for a private school to do it.
There are private schools that do try to be secular and themed on culture, I have a co-worker that sends her children to an Armenian centric school that tries to instill pride in being Armenian, etc. and the tuition is only $700 per semester. Still, she has 2 kids going to the school, so it's still $2800 per year, and that's not including all the other expense that any kid has just going to school would have, like school supplies, etc.