Elementary School - Private or state? Does it matter at this level where you send your kid?

The private schools where celebrities and presidents send their children aren’t your everyday private schools. And, I imagine that other factors play into their decisions besides the quality of education. For example, I believe Chelsea Clinton went to public school in Arkansas and was placed in Sidwell Friends partly due to security reasons. Amy Carter went to public school in Georgia and DC. The Bush twins attended public elementary school when they lived in Dallas. Now, for us “common” people, there are good and bad private schools and there are good and bad public schools. There are plenty of kids in my hometown who go to public schools all the way and end up at top colleges.