The first Heisman Trophy winner was from the U of Chicago, Jay Berwanger.
Carnegie Tech (now Carnegie Mellon) also used to be a football powerhouse, as was Fordham too.
I would argue the Cal and Stanford have a much longer and deeper football tradition of any of the schools mentioned above.
Stanford is surprising insofar as it generally fields a football team wherein at least half the players are actually students. California had for years, and may still have, a sort of three percent rule that allowed any football palooka to be put on scholarship. In either case, neither Cal nor Stanford can fill its stadiums with any regularity. Thus, no big loss.