The Big 12's stability is that every team has contracted that their media rights belong to the Big 12 for a set period of time (2022ish, I think) whether or not that team is in the Big 12. Not many conferences would be interested in a team if they can't bring their media rights with them. It would be like B1G team going to the SEC but all their games still being played on the B1G Network.
It's called a Grant of Rights (GOR) and it can be broken, but no one wants to pull the trigger on that because breaking the GOR for one conference breaks them for all of the others (Pac-12. B1G, ACC, and Big XII have them--SEC does not), and then it becomes chaos again.
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As for Oklahoma and Kansas going to the B1G, they've already been vetted by the B1G. There was an article in the OWH that quoted a current Big XII AD (anonymously) that mentioned that there was originally going to be a five team exodus to the B1G until aTm got pissy with how the money was divided and looked to the SEC instead. Five teams would have given the B1G 16, with two eight team divisions, which is what they were wanting for some time.
Also, Oklahoma's AD has gone on record with saying that the Big XII expansion needs to involve quality schools or they won't be happy. What 'quality' means or what will happen if they're not happy is nebulous...but they know they have a standing invite with the B1G if they want to pull that trigger. Plus, the school president is all about improving the academic cache of the university--something they can't do in the Big XII, nor the SEC, and something that
has happened with DoNU's inclusion into the B1G (in spite of the AAU keruffle).
As for Kansas...hell, they'll take any invite out of the Big XII. They're not choosy...remember, they were going to shack up with the Big East before that collapse happened. That, and Delaney is all about college basketball and making the B1G the perennial collegiate basketball power. Adding Kansas would do that.