The fact that Bronco emphatically represents the sponsoring Church as the football coach of its flagship university does not mean that he wouldn't adjust to the demands of a different employer with a different mission. BYU is unique. Bronco plays his role and emphasizes that uniqueness.

If Bronco were to coach at Florida or Michigan, the missions of those universities aren't precisely the same and you could expect his public persona to reflect that. At BYU Bronco wears the Church on his sleeve...because he literally wears the Church on his gameday uniform. BYU represents the Church. So he does as well.

If he were at Notre Dame, he'd represent the ideals of that institution. He would try to act well his part, because he's an honorable guy and that's what you do.