I think going independent is a big mistake and Notre Dame's slump is a perfect illustration of the reason why.
You have a program like ND, one of the greatest in the history of college football, but the game has changed. Being Independent worked for them back when they were the biggest show around (only ones getting consistent national tv time etc etc). Now that the popularity and exposure of college programs have picked up ND has fallen from the forefront
#1 Their schedule gives them nothing to play for aside from National Championships. Their last title was in 1988..their are plenty of big time programs with a similar drought or that have come out of a similar drought. The difference is those schools still had things to play for, conference titles and what not. When a recruit goes to visit a place like Ohio State they can get a glimpse of our numerous Big Ten championship trophies and all of that jazz. At ND the trophy case needs to be dusted off a bit. Notre Dame can hope to win big bowl games..but that leads to another issue
#2 Their schedule is garbage. It usually leads them to having an inflated Win/Loss record that ends up getting them matched with superior teams in Bowl games. It's not just the S.O.S. though..it's the way that it's front loaded. They usually play some good games early on against Big Ten teams but when it comes to October/November they are stuck playing nobodies (with the exception of 1 game against USC) when their are all sorts of other great games on that have huge conference/national championship implications. The fact of the matter is, Notre Dame is largely irrelevant for most of the season. Nobody aside from the die hard Domers are going to watch ND/Tulsa over Florida/LSU, Ohio State/Penn State etc.
That is going to be a problem that BYU has as well. Sure an independent sounds great if you don't give it too much thought (Hey we can schedule whoever we want!). This isn't NCAA football on 360 where you can just plug anyone into your schedule though, good teams aren't going to want to try to work BYU into their schedule in the middle of conference play, they'll be stuck playing even bigger nobodies then ND does (because hey, at least some smaller programs still bend over backwards at the perceived exposure of playing ND). BYU also won't have the cushy deal with the BCS to get a berth like ND does and that S.O.S. isn't going to get love from anyone.
:p