I have never bought that, on the whole, BYU sports is a great missionary tool. There have certainly been many instances that it was. But the nature of competition generally brings out the worst in us. Why do so many people in the old WAC region say they "hate" BYU? In most cases, they formed their hatred as a result of sports.
If BYU gets into the Big 12, there will be some missionary benefits. And there will also be a whole bunch of Texans, Kansans (is that a word?), and Oklahomans who come to hate BYU, especially if we ever come to dominate the conference.