I know we're supposed to be minding our Ps and Qs, so I want to take care to phrase this in a way that BYU fans will find religiously inoffensive.
For good or for ill, depending on your perspective, you cannot divorce that school's athletic program from the Church's mission to grow its membership. If you read between the lines, you'll see that is the driving force behind the independence move. They don't want to be in anyone's shadow, they don't want to compromise in the interest of any conference's greater good, and they're not even concerned about maximizing the program's potential for success (they are searching for the right balance of exposure versus competitiveness vs athletic respect). They want prominent media positioning for their Stanford-WestPoint-Top25-NotreDame hybrids so they can be exhibited to the greatest number of people in a positive light.
Their motives and attitudes will never make sense to the broader sports world, because they are truly an example of the sports program being subsidiary to a mission that is not even exactly educational (though providing a marketable degree is of course a big deal for BYU), but is largely evangelical.