In my experience, families with “old money” tend to be more practical and raise their kids to be humble. It’s the newer money parents who are braggy and their kids are entitled. I know that’s a vast generalization but it’s held up for many years for me. My friends in college whose families had a LOT of money were much more relaxed and friendly and inclusive than the kids whose parents were the first in their families to have money. I still see this today in our town. I mention this because it’s my guess that you’re going to find both types at any school.
I think the only way to know if a school feels more “country club” is to visit.