Call me crazy, but what if you just funded all public schools sufficiently that the advantage of going to the "good" school was lessened? Is that too socialist?
At a basic level you need teachers who care, books and decent accommodations. If those basics aren't being met, then spending will help. And if those needs are met, it certainly helps to have really great, special teachers who push students a little further.
But I believe if you took the kids from a selective school and swapped them with kids from a school with frequent violence and truancy, the kids from the selective school would still have far better life outcomes, and find ways to succeed despite run down classrooms, less inspiring teachers, and dated books. They would succeed if you put them in an 1800s one-room schoolhouse, or didn't even send them to school at all. They had to pass a high bar for intellectual ability and their parents care about education.