My take is that there indeed was anti-Catholic bias by at least some in the Big Ten back in the 1930-40s, lead by Fielding Yost, head football coach and AD of Michigan. So Notre Dame could not join the Big Ten then. Naturally, there was still bitterness after Yost died and the Big Ten were in a position to accept Notre Dame. So Notre Dame remained independent, and it became part of their identity. As time went on, it wasn’t feasible to remain independent in most sports, but remained independent in football.
Today they are independent in name only with football. The ghost of Fielding Yost is long gone. The Big Ten has pursued Notre Dame a couple of times since the 1990s. Notre Dame faculty and administration warmed up to the idea, but not so much the fans and influential donors. While there are some that still harbor bitterness that is silly at this point, a lot of fans simply hold on to the tradition, identity, and that Notre Dame has fared well in football and financially. When the last vestige of independence no longer serves Notre Dame, they will go full in with the Big Ten or some other conference.