Football and a Missionary Focus
To me it is far from clear whether the BYU football program leads to, say, more convert baptisms (and I'm not really sure of how this could be feasibly gauged). I'll acknowledge that the program may well create good feeling toward the Church among some, but having a prominent football program at a university known for its religious scruples can actually present some serious challenges that can undermine the program as an emissary for both the Church and the school. (See examples of these in these articles from USA Today, ESPN, Daily Herald, Deseret News, and more.) When situations like this arise, some of them extremely serious, the news is often widely disseminated to the disadvantage of everyone associated with university.
One might argue that the missionary work being done goes beyond just convert baptisms and encompasses outreach to many young members of the LDS Church who are positively influenced by the BYU football program and for whom many of its team members have become role models (for example, the program is an effort to combat the deleterious effects of contemporary culture on the youth of the Church). But, again, for every Steve Young, Chad Lewis, or Austin Collie, there is a Jim McMahon, Ronny Jenkins, or Owen Pochman.
I think these challenges are compounded by the pressures BYU faces in trying to recruit top tier prospects at a time when Latter-day Saint players are increasingly attending schools across the country rather than heading automatically to BYU, as they typically had in the past. I want to make clear that I have enormous respect for Bronco Mendenhall, but I have to assume the temptation to ease the "character" bar for recruits is ever-present in the program's on-going effort to remain competitive (something that has likely contributed to some of the past embarrassments cited above). Additionally, whether deserving or not, the BYU football program and its fans have gained a reputation in some quarters for being self-righteous, pretentious, and at times unruly; all of which, when combined with the success BYU has seen over the years, has helped to make BYU the team that some just love to hate.
Considering all of this and the intense passion that is invariably a part of the college football game, does a successful BYU football program really open more doors or result in more missionary referrals than would be opened if the energy and resources that are behind the program were re-directed elsewhere? Certainly there is prestige that comes with fielding a good team filled with good Latter-day Saint players (some of whom may even make it to the NFL), but isn't there prestige in fielding top academics, musicians, historians, writers, or linguists in various fields (and if not, shouldn't there be)?