Not sure what Dennis Dodd is smoking when he says that only an undefeated Notre Dame team would play on New Year's Day. The headline is a little less strict, claiming that Notre Dame needs to be undefeated to make the 4-team playoff, but even that's not true. A 1-loss Notre Dame has an excellent chance of being in the playoff.
This year there's no doubt a 1-loss Notre Dame team would play on New Year's Day, and even a 2-loss Irish team might make it this year. Dodd notes that in the future, due to Power Five conference committments to the rotating bowl schedule there will only be one "at-large" slot open among the six New Year's Day bowls in 2016. But assuming a 1-loss Notre Dame doesn't make the playoff, then 11 slots will be filled with Power Five conference teams. That's certain to wipe out all of the Power Five teams will zero losses or 1 loss, as well as several 2-loss teams. Notre Dame would be competing for that 12th spot with 3-loss teams; they would get it.
Dodd's discussion applies to BYU, not Notre Dame. A 1-loss BYU, if they continue to schedule the way they did this year, won't likely make a New Year's Day bowl now or in the future. Going undefeated is the only way they'll get there. So everything Dodd says applies to the Cougars but not the Irish, whose schedule he lumps in with the Cougars' schedule, even though they're not even close.
Notre Dame plays 10 Power Five conference teams while BYU plays just three. Notre Dame plays Florida State as well as Stanford, USC, and Arizona State from the Pac-12, all currently ranked teams. Dodd mentions that Notre Dame's schedule ranks 38th in the nation while the SEC teams are at the top, but that's for the three games played so far. Most SEC teams have played a conference game already so their SOS is high for the early season when most teams haven't played a conference game. Notre Dame's SOS for the full year is #4 according to the NCAA; that may change, but by the time the College Football Committee sets the New Year's bowl games the Irish will not be #38 in SOS.