There need to be disincentives for cheating. Financial penalties need to affect the assistant coaches too, not just the head coaches. Show-cause penalties for egregious offenders would probably curb a lot, as would disassociation penalties for "bag-men" and their ilk. A five-year ban from the NC playoff, as well as the current penalties: post-season ineligibility, scholarship losses, invalidating wins and titles, would likely limit "lack of institutional control." As would ineligibility for conference post-season revenue. Penalizing athletes for accepting cash, even while in high school, by prohibition form post-season play (for at least 3 years), added to repayment penalties, would probably deter much of that.
Subpoena power is needed. Otherwise, schools subject to public records requests get treated differently than private schools or schools in states without such laws. Being subject to subpoenas should be a requirement to play in the NCAA, both for institutions and for people (athletes, coaches, administrators, possibly even media folks).