First we had the Hollywood parents crookedly buying their kids' entrance to elite colleges/universities. More recently, we learned of wealthy kids declaring themselves financially independent in order to be "poor" and get scholarship aid. And we had people writing that higher ed is "greedy."
I have a Swiftian 'solution' which will solve some of this dilemma. Elite higher ed institutions, those with many more applicants than vacancies, should set aside perhaps 5-10% of the freshman class slots for an auction.
What I propose is they sell these slots off to the highest bidder, and do it quite openly as a way to fund other good things the institution would do. I'd suggest an opening bid of $500,000.
The hyper-wealthy can openly bid for a slot for their child and fund higher ed at the same time. If the institutions want to use the monies so raised to fund support for bright poor kids, fine. If they want to use it to hire an excellent football coach, equally fine.
But perhaps you say this would devalue degrees earned at that campus. I'd respond, we're already admitting student athletes and affirmative action recruits with merely adequate minds, why not a handful of ho-hum rich kids. They, like the other two categories mentioned, bring a substantial benefit to the campus, one that transcends their mere presence.