BYU athletics are NOT a missionary tool.
Explanation for #1: BYU does not recruit well enough in the modern era to compete with the likes of Alabama, Ohio State, USC, Florida State, et. al. They will NEVER recruit that well, and will never have the talent or depth necessary to compete at that level.
Explanation for #2: I'm not saying that BYU doesn't think it is, or that you don't think it is... I'm saying it isn't whether they or you think so or not. I haven't met a person yet who, when watching BYU football, said gee I think I'll find out what the mormon church is all about now because I'm watching BYU play. You might be the one or two in a million who has a different story to this, but that doesn't constitute a missionary tool. The money would be better spent in actual missionary development then in athletics if that is what they were going for... I don't think it is.