You think that if half our players were not LDS we would be making a bigger missionary impact with football? I don't see how that would be true. Odds of HC violations would go up. We would have issues all the time. The media would either emphasize the "strange" rules and rather than normalizing the church, the football program would only further marginalize us in society, or, if the violations were not acceptable in society either, then they would associate the players actions with the church, member or not. This is because players represent the church in many nonmembers eyes.
If, on the other hand, player obedience would be just as good or better, then, at best, nothing changes.
The only argument I see working for you, is that BYU footballs impact on society as a missionary tool is nonexistent, in which case, we might as well try to convert a few football players.