I'll say the opposite - all college donations and income should be publicly available as soon as the ACH transaction clears. Questionable sources immediately flagged. Make every single dollar within the education system completely transparent. Then if some kid gets admitted, and we also see their parents made a $5 million donation, we can go ahead and appropriately stain that kid's admission.
What about the kids who really earned that admittance and their parents made a sizable donation? Then the parents donate their funds to another educational institution. Or the kid attends another institution. Wherever these two actions overlap - immediately assume fraud over anything else.
On bribes directly to individuals - their funds are all publicly available as well. If that's a problem, there are millions of other jobs they could pursue rather than being a useless parasite on the education system.