to answer those who joke about conferences having nothing to do with academics, on the surface, sure, it seems crazy, but Presidents are the decision makers. Presidents want to align with other Presidents with similar goals. Athletics is nothing but a marketing division of a major university, so think of athletics conferences as university marketing partnerships. some Presidents would prefer not to align their marketing efforts with schools that don't align with their academic profile/mission.
L'ville, i believe, was a decision made in fear of the ACC falling apart. ACC Presidents were willing to "sacrifice" the overall profile of the league in an effort to keep the existing partnerships intact.
all that said, if a university's athletic department was put in a position of being irrelevant, the President would have to decide what to do, and would likely seek a path of relevance since athletics is the face of the university. that is why people say the most presidents in the ACC want it to remain intact, but if it begins to fall apart, it would ultimately be every man for themselves.
going full circle in this thread back to the AAU question. it's simple, AAU schools like to be affiliated with one another (including through marketing/sports), and schools aspiring to be AAU would be doing themselves a favor by aligning themselves as closely as possible with other AAU schools. Sports may not be the only way, but it's a really good one since athletics conference presidents are very active with one another.