Simply put, the old system like you mention, with a "Plus 1 as needed" caveat. In a situation like 2004, USC would play Auburn for the NC.
More thoroughly, I'd like to go back to early 90s conferences of 10 teams each and equalize things in terms of number of teams per conference and number of conference games for all.
The ACC would have to add, the Big East would include Penn St and ND, and so on.
An equal-footing, regional system of conferences, with no RB tie-in screwing it all up (1991, 1997, etc).
No big, fat lie necessary. Whether FBS, upper Division 1 is comprised of 80 or 66 or 40 or whatever number of teams, it's balanced and equal. Traditional bowls work, but we'd all want a 1 vs 2 matchup. Traditional bowl tie-ins after that are fine, if the conferences want them. Or keep it as it was back then, but having a site for a +1 as necessary (1994) as well.
The key is not pretending JMU and Ohio State are playing the same game. OSU is Formula 1 and JMU is a kid in a go-kart with a lawn-mower engine on it. Just stop it.
If Rice is still in the SWC with Texas, then there are no built-in obstacles to Rice winning the SWC and vying for a NC. If they're a cellar-dweller for 40 years, that's on them, but at least the format doesn't make them a 2nd-class citizen, their actions or lack of actions does.
G5 or mid-major programs are either in the group of conferences in this division or they're not. If no conference picks you up, then you're out. Go be in your own division and matter there, instead of being irrelevant like you are now.