This is, I think, a misapplication of Berkson's paradox. The assumption is that, all else equal, a university is less likely to accept a public school student.

This is, I think, a misapplication of Berkson's paradox. The assumption is that, all else equal, a university is less likely to accept a public school student. But there's so many confounders here that Berkson's paradox is unlikely to follow trivially. This might be true, but I expect it's more true for (and as a consequence of) parental income/legacyness than than preference for private schools.