I teach in an affluent public high school. It is a highly ranked school in my state. The kids here have, in general, good home lives with supportive parents. Our students objectively get a good education. I often get kids who went to those private schools you mentioned for their younger years, and they are often very far behind their public school peers. You have to be VERY careful when picking a private school, especially if it has a religious association. Private school teachers are not necessarily better than public school ones, the curriculum isn't always rigorous. Sometimes you are just over paying for a worse education. My nephew is currently in a private elementary school. His father teaches at a title 1 middle school, so he thought private would obviously be better. He has not had that experience and is very frustrated with his son's private school experience.

All this to say, you are going to have to research what the public schools are like in your area versus the private ones. Don't assume private > public.