Now that I have a tool that can report on how much controversy any given scheme would have, I can start looking at numbers!
First, I have to decide the range of seasons to check. The earliest season I can use is 1872. See the writeup on the code for a discussion of why 1869 through 1871 can't be used. An argument could be made that I should start with "the modern era", whatever that is. Arguments can be made for years as early as 1906 and as late as 1985 - and many in between. I'm going to take the "more data is better" approach, and start with 1872.
Likewise, the last year to use is open to debate. Here, the issue is bowl games. Ideally, the data would use records from the regular season and possible conference championships, which are played the last day of the regular season. Unfortunately, the data source also includes bowl games. Before 1992, the bowl games were no better organized than the regular season. Conferences and bowls had contracts for teams to play in specific games. So you would get a selection of highly ranked teams playing each other apparently randomly, with little chance of the two highest ranked teams playing each other. While these games may well have changed which systems would have been controversial each year, I believe the effect will average out over the long term.
This changed in 1992, when various entities started trying to get the best two teams playing each other. There was limited success until 1999, when the current system - agreed to by the major conferences - was started. Because the system matched the top two teams in 1992, I'm going to stop with the 1991 season.
Majors vs. mid-majors
One of the controversial features of the current system is that the FBS schools are split into two groups, "majors" and "mid-majors". The major (aka AQ or BCS) schools get ranked above the mid-majors, even if they have more losses. The explanation for this is that schools in each group mostly plays schools in that group, so mid-major schools don't play major schools very often, and hence have an easier schedule. I purposely ignored that, because an undefeated mid-major team causes controversy, especially if there are no undefeated major teams. Again, the controversy the code finds may be different from the controversy the press sees, and again I expect that to average out over the long run.
And the numbers are
As you can see, there isn't a lot of difference to be seen between the three options. The championship game we have now seems to generate slightly less controversy, which is what we've experienced. Going to a plus-1 from what we have now would raise the controversy level back to almost the point we were at before.
Our current system with a single championship game is less likely to generate controversy than going to a plus-1 system.