For me it's less embarrassed to be associated with certain schools but more of an "island of misfit toys" conferences are doomed pragmatism. The old Big East was a total mishmash of schools that had nothing in common on the football side of things and then you had the basketball schools and all of their needs, every deal seemed to be based on what's good for Providence and Seton Hall. The ACC was an improvement but still had the likes of BC and Wake, the good old boy network of bs like the Raycom deal, all the power being located within the four North Carolina schools, why do you think Maryland went into severe debt when they had a chance to leave. Also the SMU addition was dumb, the whole "they're rich" argument is silly and it just opens the doors for more "they're rich" schools to be the ones added once the B1G and SEC raids happen.
A brand new conference of good to great public schools, blue blood private schools like Duke and Stanford and then a nonsectarian private school that's bigger than some public schools like Syracuse, is the way to go. There's way way too much baggage with small private schools no matter how much money they have, little brother state schools, religious schools with tons of rules on when they can play, and religious schools that have histories of covering up crimes.