That’s true in a sense. Money is part of the game. All teams pay each other. The way it works is that the two teams share in the proceeds from the game, gate receipts, tv and radio revenues, each getting an agreed upon percentage. Some teams have a bigger stadium and hence larger gate receipts and some teams have a big national following so tv and radio revenues are greater. Hence, since the teams share in the game revenues, it is desirable to play a team that will generate a lot of revenue from a game.
A small school may have to coax a big name school to play them by guaranteeing the big name school a certain amount. That is, why should a big name team play a small team that only has a small pot to share? Unless they are guaranteed a minimum payout.
This is true whether the teams are independent or not.
Of course conference teams have to play the other conference teams, but the pot is shared as well. And with the conference.
And bowl games simply are all about money. Teams are paid to play in a bowl game, and some bowls pay a lot more than others. Bowl games are run privately to make money, so they invite teams that will make money for them. Of course there are contracts and agreements between the different bowls and the different conferences. For instance, the Rose Bowl would invite the champion of the Big Ten to play the champion of the Pac 10. Other bowls and conferences have similar agreements.
Independents like Notre Dame therefore rely upon at-large bowl bids. However, not to worry, because what bowl committee with an at-large slot to fill would not be glad to have Notre Dame in it!
I guess BYU thinks it can go that route too.