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With your clarification, my point does beg that... by Kbyrnes

...question, in a post hoc ergo propter hoc manner, I think--but, whatever. By your clarification I refer to this:

"My objective is broader than that -- it is to crown as champion the team with the best performance over the course of the entire season."

This is what we had when national champions were named before the bowl games. The voters used whatever subjective and notionally objective criteria they thought were appropriate to vote for #1 at the end of each regular season. You often heard that a major criterion was a team's "body of work" for the season; that also came up when we were in the season of selecting teams for the BCS, and it's persisted into the playoff era.

A playoff system inherently contradicts the "body of work" concept to some degree; your "body of work" presumably allowed you to get into the playoffs, but once you're in, that body of work became irrelevant. Just win all your playoff games.

Several sports keep track of who is the "world number one" via various point systems. In golf, they have a point system for events and you divide your points won by the number of events you were in; whoever has the highest average in points is the world #1. You don't directly win anything for being #1, other than prestige; there's an award for most weeks in a year at #1, but no money attached to it. In chess, there's the ELO ranking, and at any given time you can look up who's #1 in that system--currently it's Magnus Carlsen.

For college football, we already have several ranking systems: Sagarin et al. At the end of the regular season, you could pick which of these systems you think was the best, and then, seeing who they had as #1, get the answer to your search for "the team with the best performance over the course of the entire season." But did you mean regular season, or the whole season including playoffs?

Your 4th paragraph confuses me a bit. It seems as if the point of your single elimination playoff would be to validate a team that we already thought could have been the best. This seems to have been inferred when you wrote, "If a head to head matchup would not resolve that question in favor of the victor, then such a team shouldn't be included in the playoff."

What do you think of the NCAA basketball postseason? The top seeds are almost always teams that would fit your concept of a handful of top teams that would fall within the bounds of reasonable disagreement. But then there are all the lower seeds--each of which has a theoretical chance to knock off a higher seed and eventually advance to the finals. The NCAA typically seeds 64 teams, which is a lot more than 12--do you see that as a flawed system?