Of course they can but will they is the question.
Couple of years ago, Harvard mens soccer team was caught rating the women soccer team players' attractiveness, and did so by having a website for it. Incredibly stupid but nothing compared to what has happened to the scandals herein. That said, they forfeited all of their games and ineligible for the NCAA playoffs. They were the Ivy League leader at the time in standings. One of their assistant coach left as a result, even though he had nothing to do with the problem, other than he was in charge of recruiting for Harvard mens soccer. The character of the players were the issue and they probably needed a fall guy.
Each organization, private or public, has a choice how they decide to deal with crisis situation of this kind. Often Tylenol example is used as a landmark "right thing to do". If one apply the similar stance, any university should fire their athletic director and all of the coaching staff in that sport, and suspend the program for a period of time that takes to fully investigate and put remedy in place - it happened under their watch. Clearly, this would not be fair to all the players but the university has higher integrity issue and the school needs to help place the players onto other programs/schools, if they so desire to play.
As I've watched the behavior of USC and UCLA regarding this matter, we have decided our youngest kid (boy) who is a junior in HS, will not apply to either of these schools. He has the grades (GPA 4.7, SAT 1490) but we'll send him elsewhere. He plays soccer but does not want to play in college, like his brother is doing. Taking the recruiting element out makes the school choice much easier.
At the end of the day, quality of education can be found in many. many places. We raised our kids to believe in integrity and honesty, and act accordingly. It would be in direct conflict to send him to one of these schools, given how they've dealt with this particular scandal. In particular to USC, there are too many other examples of not dealing with it quickly and directly that we would not consider it amongst private institutions we are looking, inspire of their excellent alumni network.