I’m not ignoring the difference between any elite high schools. They clearly have a major edge in ivy admissions. I went to BxSci. Many classmates went to Brown, Cornell and Columbia but it was a different time. The top kids went to HPM if they could afford it. Many did go to SUNY (Albany,Bing, & Buff engineering back then) for financial reasons.
Where is gets more interesting are not not so elite but good & expensive privates. (not talking about local religious schools). We have a bunch of these in my area and they have very nice Ivy results. Actually the public with a significant volume of Hi-stats ORM kids have a much better T10 rate than our similar school.
You know we are pretty much in agreement overall. We differ on how much non-elites matter. If I had access to all Ivy applications for the past few years, emails from high school counselors, emails from the development and alumni offices and a list of phone calls to admissions with a few grad students to do the grunt work, we could answer the question.