Anecdotal, but… our experience is yes. S attended a public school, did well and attends a private highly selective college. However, most of his class (and previous and future classes - like overwhelmingly most) attend state U. A few neighborhood kids attended prominent private schools. They and many many many of their peers attend quality private schools. Like almost all of them. All over the country too.
I recall when S was applying to schools, the really high end ones all had local info sessions at his friend’s private schools. They were open to basically anyone, but the fact that they were on the private school’s campus clearly shows their relationship. Typically these high schools send kids to top colleges each yr and there is a certain expectation on both parties as they have a known “product” and success factor. Colleges want to admit kids that will have success, graduate on time, do well, get a good job, contribute funding, etc. If there’s a long term track record in place, it makes it easy for the college to swim in that admission pond.
But as evidenced by my own kid, as long as you are a strong candidate, you can get into these great schools from anywhere.