While the title is "the solution", this is, of course, my proposed solution. The problem to be solved is that the current system leaving "deserving" teams out of the playoff creates controversies.
Why a playoff is not the solution
A playoff would seem to be the obvious solution, but there are some real reasons not to change. Ignoring the financial interests, a real playoff system - whether it's 8 teams (the major conference champions plus wild cards) or 16 (all FBS conference champions plus wild cards) - changes the nature of the regular season. The more teams allowed into the playoffs, the less important - and hence less intense and less exciting - the regular season becomes.
While it's now a cliche, with the current system every game counts. If a team is in contention for a national championship, losing any game changes the nature of their season. It may be enough to take them out of the national championship picture. At the very least, they'll need help - in the form of other teams beating contending teams - in order to have a shot at the national championship.
The conference championship is a different thing entirely. The non-conference games aren't counted in conference standings - or only counted indirectly, in that the rankings are used as last resort tie-breakers, and the non-conference games count there. Losing a game to a conference opponent that's already got more losses than you doesnt't - by itself - change the nature of your season. If you win the rest of the conference games, you'll be tied with someone that you beat and hence hold the tiebreaker.
So a team can be completely out of the national championship race and still be a contender for a conference championship. A playoff system changes that because if you're in contention for a conference title, that puts you in contention for a national championship. This makes the regular season nothing more than a qualifier for the playoffs, and much less exciting.
The Flex Playoff
The goal is to preserve the "every game matters" nature of the regular season, while at the same time avoiding controversies when it comes to picking teams for the playoff.
The solution - to me, at least - seems obvious: let the number of teams in the playoff change, so that each year only the "deserving" teams are included. This means the format changes each year, but the bowl system is flexible enough to accommodate that.
To do this, take the top two teams in the rankings, and anyone with no more losses than the number two team, seeded by ranking. That eliminates controversies - at least so far as choosing the teams is concerned. It means that a team that loses a game needs help - in the form of all but one other undefeated team losing - to have a chance to get into the national championship game. It means the bias against the mid-majors will only lower their seeding; they'll get a chance to prove they deserve better on the field.
This is still using the regular season as a qualifier for the playoffs, but the level to qualify is the nearly the same as it is now, so the regular season is nearly as important as it is now. The format would be the same as an 8-team tournanemt, with teams facing a non-qualifying seed (i.e. seed #5 when only three teams qualified) get a bye. With fewer than 5 qualifiers, the first round is skipped and there's a four team tournamet. With only two qualifers, there's just a championship game.
So a flex playoff will have all the fairness desired of a regular playoff, in that no deserving team will be left out. It will be more fair than a playoff, in that no undeserving team will be allowed in. It will maintain the intensity of the regular season at the level it is now, because qualifying for a flex playoff requires the same record as qualifying for the championship game. It combines the best features of both systems.