SALT LAKE CITY—Everything was going right for Utah’s football team. The Utes had just clinched their second consecutive Pac-12 championship appearance, and they were about to learn how much closer they were to the College Football Playoff when coach Kyle Whittingham gathered his players last Saturday. And then he mentioned another team.

It wasn’t Colorado, the team his Utes had just stomped. It wasn’t Oregon, his team’s opponent in the Pac-12 Championship on Friday. It was a school on the other side of the country that Utah...