Utah will be better off than those tiny private schools. There's less pressure to succeed at those schools because they're valued for their academic prestige. Stanford's current run is an anomaly. Even Northwestern had a pretty good run under Gary Barnett.
But I don't see why Utah should be any better than Minnesota or Iowa State. Minnesota's an especially fair comparison. Good "research" school with large fanbase thanks to soft undergrad standards. Medium sized metro setting in a cold weather state. Major conference affiliation.