ND has a standing invite to the Big Ten only for all sports. If ND football joins the Big Ten, they become largely a second tier football program, under OSU and Michigan, in most years.
Notre Dame has certainly built itself a legacy, and it has a nice carve-out as the only "P5" Independent, but they have hidden behind that legacy for about 25-30 years. They've preformed fairly well in most regular seasons, and even beat some highly ranked teams, but when the rubber meets the road, they have a tendency of getting rolled.