most students at harvard and other elite universities come from american public schools, for the record. very high concentration of rich preppy private school kids, yes, but majority is still public.
How many of those public school students are coming from exclusive suburban districts that may as well be private given the cost of living within the district's boundaries? I would guess the majority. Sure the random star from small-town Illinois will get in alongside 20 kids from a Chicago suburban high school.
That would be my thought as well. I'm originally from NZ so there's negligible private schooling, the curriculum is standardized, and obviously the country does a much better job at looking after people on lower rungs of the social ladder.
Public schools in NZ are also forbidden from what NZ calls "zoning" which I think in the US is explicit and mandatory through school districts - e.g. the exact opposite.
Despite that, public schools get some reasonable portion of their funding from the council/city/region/district/etc so a public school in a rich neighborhood generally has more funding. The next biggest source of funding is community/parent groups doing fund raising and what not - school fund raisers are common in NZ - but that again means a school in a richer area ends up having more resources.
The end result is that even in a country where there's negligible private school, schools are explicitly prohibited from restricting enrollment geographically, there's a significant difference in outcomes for rich neighborhood school vs poor.