Entirely wrong.
In a new Big 12 with a 7 game conference schedule and Air Force in the conference...ND could still play Michigan, Michigan State, Pitt, Stanford and a rotation of Purdue-Navy-Army....while playing the other Academy every year. So ND's rivalries will be intact. PLUS they get their own TV network....in other words, they get every benefit of Independence plus they can add Oklahoma and Texas to their schedule every year. AND...they would be the lead team in the NOrth...with a likely conference championship game most seasons.
Its all upside for ND...and most likely terrific payout when you include a Texas style ND network deal.
Well, the Big 12 is not going to a 7 game conference schedule. But in any case, lets look at the formula Notre Dame follows in its scheduling:
3 Midwestern teams
3 Eastern teams
2-3 Pacific coast teams
2 Service academies
1-2 Random others
Notre Dame's schedule in an expanded Big 12 (and I'll be nice and make it an 8 game conference schedule instead of 9):
8 Central corridor teams
1 Pacific team
1 Eastern team
1 Service academy (Navy for historical reasons)
1 Midwestern team
That is an evisceration of the identity of Notre Dame. They currently almost never play central corridor teams, because there is little interest in Notre Dame football in that part of the country. Virtually the only team they would want to play would be Texas, and they can schedule Texas now, or if they joined the Big Ten or Big East.
And here's the thing, they could come far closer to playing a Notre Dame schedule in either the Big Ten or the Big East, which is why, were they to join a conference, it would be one of those two. Joining the Big Ten they could keep Mich/MichSt/Purdue/Penn State on the schedule. Add their non-con games and they could end up with a schedule like the following:
7 Midwest
2 Eastern (inc Penn State)
1-2 Service academy
1-2 Pacific
Its still a big change, but not the end of their historical identity, and keeps them in the part of the country (the Midwest/Northeast) in which they are highly relevant.
Notre Dame wants to stay Notre Dame. That's why its not in a conference, but its also why it would never-ever-ever join the Big 12, and could - in extraordinary circumstances - join the Big Ten or Big East.