At the risk of going too hard against the grain here, I’ve always found this sort of mentality to be a bit off. Given the high monetary cost, the factor of being away from home, etc., I certainly would have not attended a private school had I not gotten into the one I really wanted to. Some people apply to schools like E/A because they want to attend a specific school, not just boarding school in general, and the “expand your base” mentality simply isn’t right for everyone. (Whether this applies to OP or not is a different situation).