young Holden doesn't so much flunk out as he's *expelled* over and over because he's a total fuck-up.
It's my impression that back then academics often didn't matter very much in the private schools. I could be wrong. I do know that good secular private schools these days have a very strong emphasis on academics, and it's somewhat harder for the rich, privileged kids to skate by.
Even so, the world of private schools, even today, and I think a lot more so back when Salinger was writing, is a world in which there aren't really any consequences. That's still true today with rich kids. They can screw up over and over, get bailed out over and over, and never quite get it that there will come a time when they won't be bailed out. As I noted above, I witnessed that at my kids' private school.
Not that the parents of public school kids don't also do that, rescue their kids and protect them from consequences, far too often.
Holden's wealth, actually his family's wealth, is important, because without that, he'd have been a high-school drop-out, would have gone to work in a factory somewhere, become an alcoholic and an abusive husband and father.