I've watched some Utah games - or at least portions of them. And you and I are seeing very different things.
I have yet to see a Utah home game where they talk about pioneers, or Brigham Young, or where they've told the audience that Kyle Whittingham is LDS. I have heard them talk about players being RM's before - so there is some of that. (OK - I suppose if I think really really hard, they may have mentioned that KW attended BYU - but that's as close as I've heard them come to saying KW is LDS or even mentioning Brigham Young by name.)
I think I've seen one shot of the temple. One shot once.
And I get that you're saying they stay out of the news as a good thing related to NCAA violations, etc. But how can you draw attention to something - like the church - if you're out of the news? Seems silly to me. Frankly - while BYU has been in the news for Emery and Childs lately - in both cases the local and national news both have said that the consequences to BYU are ridiculous and overblown - and they've all said that the university was cleared of any wrongdoing in Emery's case - the coaches didn't know about it.
So - I'm trying to see where you're coming from. But I just don't see it.