The opposite: the larger the field, the more teams that almost make it, as the expanded NCAA tournament field has demonstrated over the years with more "bubble teams" and more disputes.
And the major conferences would never agree to conference champs only because the disparity of competition in different conferences is so great.
Do you remember when the NCAA tournament field was only 12 or 16 teams, mostly conference champions? Far fewer disputes then than now with 64+. There were years and years where the only SEC team to make the tournament was Kentucky, but no disputes since only one could go, and the basketball talent was more evenly spread than football talent.
No system will end disputes, they just need to decide how much to expand the NCAA football championship field beyond two teams.
No “bubble team” would be able to lay claim to the national championship game under the current system, so a 16 game playoff would be an improvement over the current system. Teams ranked that low are 2-3 loss teams or 1-loss teams with an easy schedule. What would be really exciting is the final weeks of the regular season, where teams would be vying for the cutoff position!
With a 2-team playoff (as we currently have), you have #3 and sometimes #4 claiming they were unfairly excluded from a shot at the title. There have been years when there have been 3-4 undefeated teams, for example, but major conference BCS bias kept out folks like Boise St.
With a 4-team playoff, you may have some cases where #5 whines about not getting a title shot, but they really have far less claim when you consider that under the current system (or even the pre-BCS system), it's fairly rare for #4 to have a legitimate claim that they should have been considered. No, at this point, it's about schools wanting a shot at the money more so than the title. Of course a lower-ranked team can always beat a higher-ranked one, so the lure of a title is still there, but you know #5 is going to have to do it as an underdog.
As the field size expands, the situation continues in this same vein. #9 complains, but they're an even longer shot. Ditto for #17. For them, it's about qualifying for the lucrative and high-prestige tournament, not because they have any real hope of winning it. (Q.v., the NCAA men's basketball tournament -- starting around the #13 seed or so in each bracket, you're dealing with schools who are just happy to make the tournament. A #13 or #14 might surprise enough to make the second weekend, but no one lower than a #12 has ever made a serious run at the title.)
Bearing all of that in mind, a 4-team or 8-team makes the most sense. After you get past 4 teams, you're starting to really increase the chance for a team with a "failed" regular season to have a chance to win the championship, but an 8-team playoff would allow someone who just had one off week or some really bad luck a chance at redemption. 16 would be overkill.
(For the record, I think the NFL currently allows too many teams in its playoffs. As much as it would have been to my own team's detriment this year, I despise wild card berths -- they were a necessity when there were three divisions per conference, but it's grown into a monstrosity.)