Church leadership groups don't simply "toe the line" when it comes to decisions. They each have opinions, which they discuss, review and ultimately come to consensus on. This is why it takes so long to make decisions sometimes, they wait until the come to a full consensus, they don't simply drive things through---even when there may be a loud voice from one of the most senior leaders, they still wait until they agree.

If they were to shut down athletics at BYU, it would be because the BOT, not just 1 member of the Board, felt that it was the right, or necessary thing.