One thing I have thought about over all these years is what we could do that fans would actually buy into and rally around and that the administration would support and I am somewhat in the same mindset as you. If we were to go in that direction, I would also lean heavily on the "Fighting Illini" and "Memorial Stadium". Maybe in the pregame or halftime as part of the 3 in 1, start pouring down smoke, silence the crowd, and strike the bell, and out from the smoke steps out selected students dressed as our greatest players and figures as they were in their playing days. Red Grange, Dick Butkus, Papa Bear Halas, Ray Nitschke, Jim Grabowski, Simeon Rice, Jeff George, etc. and make some sort of pageantry of it whether it be a presentation of trophies or championships they won (yes that does sound ostentatious), or leading us in Loyalty. And also have a special honor that day for one of them where similar to the Red Grange run you have a fan challenge them between quarters (imagine for Butkus, a fan and the Butkus impersonator in those inflatable balls and the fan has to get past Butkus in 10yds for the first down. Fans would devour that).
Basically don't give a single mascot, but instead almost make it a bunch of our former players as technically mascots to honor our "Fighting Illini" and traditions. And you could do the same with basketball. The Legendary Fighting Illini. Something like that.