But let’s me add some counter example for the sake of it.

here is public high school, with « famous »teacher and kids clearly coming from the wealthy élites.

https://en.m.wikipedia.org/wiki/Lyc%C3%A9e_Henri-IV

The trick is that the selection process is so stringent that you basically have to be :

- a genious in getting excellent grade according to the French system or … - have parents that coach you to get in there.

Here is another similar public high school.

https://en.m.wikipedia.org/wiki/Lyc%C3%A9e_Louis-le-Grand

Again, I mostly agree with you point. But dépending of the country, a private education can be seen of « bad taste »