Viewed from a certain perspective, a university is a machine whose purpose is to turn high school students into donors (put more nobly, "alumni"). In the main, schools impart the character necessary to write checks by fostering a unique kind of student culture, for better or worse: Princeton students have an uncanny graciousness in all circumstances and only some awareness of what their personal limitations might consist of; Chicago students work harder than any others, but take the school's unofficial 'where fun goes to die' motto as a mirthless prophecy rather than self-satire.
Flagship public universities are not like this. They offer no one singular experience that it is possible for everyone to participate in, or shapes them whether they participate in it or not (Duke students who do not participate in the Greek scene nevertheless have their experiences shaped by that scene). My alma mater has 20,000-24,000 undergraduates floating around at any one time; my freshman dorm held 1000. There's no reason for any of these people to interact: Nursing School students spend all their time at one end of campus, apart from other academic buildings; the Art, Music, and Engineering schools are on a separate campus accessible by bus, but no less than a 30 minute walk from the next closest part of the campus (if you're willing to jump some train tracks). There are broad distributional requirements that can be fulfilled in any number of ways, and even though everyone is required to take a section of freshman comp, the sections vary as widely as the (large number of) people who teach them. We have no reason to have anything in common except occupying the same (very large) space for a period of time.
But we do have football. I've mentioned this before, but one of the details that The Big Chill gets absolutely correct is when everyone sits down to watch football. Everyone follows the football team. Not everyone goes to the games (I didn't), not everyone knows who the players are. Not everyone even entirely gets how football is played. But everyone, while you're on campus, knows whether we won or not, and (usually) at least some of what happened during the game. People who go to smaller schools tend not to understand this common social element: not everyone loves the team, but everyone follows it, and it's one of the very few things all these people have in common. We might stick to it to a greater or lesser extent in the following years, but it's (one part of) what we did when we were there.
My experience with team sports ended with two not-particularly-glorious years of junior high track. Our coach would remind us at every opportunity that whenever we wore our uniform, or our sweats, we had to be careful, because there were little kids who were looking up to us and would emulate whatever we did. I'm pretty sure I scoffed at it, even then, because what kid looks up to the junior high track team? Subsequent experience has pretty much confirmed her to be right: in college I worked at a science museum with an emphasis on children's education, and was required by the management to always be in logo-bearing shirts while working. Walking to and from that job, I would occasionally come across kids out with their parents, and without fail the kids would notice my shirt and get excited. They also would watch whatever I did, and they would try to emulate it. If I crossed a street against the signal, you could hear them asking their parents why they had to wait.
If there's anything else Michigan prides itself on, it's taking that approach to absolutely everything. I sometimes like to joke that the New Testament's ethic boils down to "everything you do matters all the time," and it's that sort of ethic that the university tries to instill. A Michigan Man will be thoughtful about everything, always cautious about what he's done and is attempting to do, and genuinely worried about identifying the right course of action, to the point of overkill. If you're a state school, this is a very sensible ethic (Carolina has its own variant on it): some of your students will go on to do stereotypically great things, but most of them will return to their communities, or find their own, and will be constantly bombarded with the expectations that come from leadership on any level. As Aristotle would have said, one practices the acts of a virtuous person so that when the time comes, and making the right decisions under pressure is needed, the right actions will appear as if by instinct.
Integrity is, as they say, what you do when no one is watching, when there's no one to lie to but God or yourself, and in that moment you get some sense of who someone is. The academic world has its own parallel to this, when someone reaches out to help you even when you could not possibly help to advance their career, and the time they take on you leaves them with less for their own work. People who are good, insomuch as anyone is good, will do these little and unnoticed things for the sheer reason that a good person does them. Michigan has its own exemplar of this character in Raoul Wallenberg, a student in the 1930s who could have taken advantage of his family's industrial connections to sit out World War II entirely, and instead created and implemented a scheme that saved the lives of tens of thousands of Jews in Hungary at the end of the war.
Integrity is also what you do when everyone is watching, and at this, Michigan has systematically failed. The football coaching and training staff did a horrible thing, allowing a concussed player to continue playing (for those unfamiliar, a second blow to the head for a concussed person can result in death). The virtuous action here would be to take responsibility, apologize, and vow to improve in the future. Instead, we have seen: denial that any head injury occurred, unwillingness to admit fault or culpability or even that a mistake was made, releasing alternate (conflicting) explanations without noticing or caring about contradictions.
Which leaves us with the question: who, basing themselves only on the words and actions of Michigan's head coach in the last week, could possibly consider him a role model? And if Michigan's coach isn't going to be a role model, why are we bothering with football in the first place?