The actual problem here is not the bribery. The problem is that those bribes are being captured by random 3rd parties, rather than by the school itself. The correct way to implement college admissions is this: Charge a sliding scale fee based on application quality, and then use the excess revenue generated to subsidize poor kids tuition. Your kid has a 2.5 GPA and wants to go to Harvard? Great, that'll be a million…
No. No amount of money should get you a way to get into an "elite" school if you don't deserve it. Unless we're ready to admit that these schools aren't really elite if any billionaire's dumb dumb kid can get in with enough money thrown at the institution. The issue with trying to solve every single problem with money is that capitalism isn't designed to solve problems where a moral compass is needed.
Do you really think this isn't already the case?