I think the guiding principle should be that if a team beats everyone that is put in front of them, they are the champs.
I would give auto-bids to each of the 10 conferences. Conferences could have the power to determine that champion by whatever way they want (regular season only, title game, tic-tac-toe), it's on them.
Then, there are six at large bids based on the top six ranked non-conf champions. Total of 16 teams in.
Seeds 1-4 get a double bye.
Seeds 5-8 get a single bye.
Seeds 9-16 get no bye.
If you are a FBS school that has committed to fund a football program, you cannot complain the system is not giving you a chance. If you are a big boy like an Alabama or Clemson, you're getting a double bye and the team you will be playing in the first round of the playoffs has had to play at least one game already, if not two.
You can add even more advantage for the better seeds by having the games at their home stadiums, whatever you want.
Independents have a shot with six at large bids. That is plenty to afford them a place at the table.