There are plenty of reasons for BYU to join the Big XII: BCS access, great opponents, and recruiting opportunities. There are also lots of reason to avoid the recently shrunken conference. For example, recent rumors have Texas A&M bolting to the SEC, leaving the future of the conference in question.
But the real reason BYU will stay independent is summed up in one word, a word that has been uttered quite a bit in Provo the last year: exposure.
BCS access has never been BYU's number one goal. BYU doesn't just want money; it has plenty of that. What does BYU want? BYU wants to travel the country and play football in front of their fans in all time zones. It wants ten games on the ESPN network. It wants to schedule a season with games against Texas, Notre Dame, Georgia Tech, Utah, Boise State, and Hawaii. That schedule includes games against the Big XII, ACC, PAC 12, Mountain West, and Notre Dame (yes, they are their own conference). By the way, that is BYU's 2013 schedule.
If BYU joins the Big XII then it is back to scheduling 3 nonconference games a year, two of which would likely include Utah and Utah State. That would mean that they would play their entire football schedule in the midwest and Utah. A season without at least one game against a west coast school doesn't seem likely and BYU would also love to get at least one east coast team on the schedule.
Fans argue that BYU can always go back to independence if the Big XII ultimately disintegrates, but independence has never been a backup plan for the Cougars. There may have been a few BCS conferences that Holmoe and company would have considered joining, but an unstable Big XII is not one of them.
BYU is independent, and that is a great place for them to stay.