There are at least two reasons not to treat Corona outbreaks at schools like cold or flu outbreaks, and TBH even during a flu outbreak, it's reasonable to expect people to get vaccinated, even if it's not required like other vaccines. In some cultures, it's even expected that if you're at all sick, even with the common cold, you'll wear a mask to protect others as well, but I get that that's not the norm in the US or around Stanford.

First, there's not just healthy "college-aged" students on campus. With that in mind, there's a risk that area hospitals could get overloaded with patients. You don't want the hospital to be full when there's a car accident or someone needs their stomach pumped, someone suspects they might have an STD, or whatever. Requiring a booster mitigates that risk as it lessens the symptoms to the point where hospitalization is shorter/unlikely.

Second, there's a chance for a mutation each time the virus spreads from one person to another. For that reason, you don't want it ripping through everyone.