Here's what the playoffs would look like if only conference champions were allowed. The rankings used and listed are the BCS ones.
No. 1 Oklahoma (12-1) vs. No. 6 Utah (12-0)
No. 2 Oregon (12-0) vs. No. 3 TCU (12-0)
No. 1 LSU (13-0) vs. No. 7 Boise State (11-1)
No. 3 Oklahoma State (11-1) vs. No. 5 Oregon (11-2)
Opponents of this plan don't have to look back very far to make their case. To me, the 2011 season serves as the worst possible chaos for the "must-win conference" scenario. In the BCS rankings that year, the No. 2, No. 4 and No. 6 teams didn't win their leagues.
At the same time, that same year does offer up an example of why it might be nice to use the scenario. According to the BCS rankings, No. 4 Stanford is ranked ahead of No. 5 Oregon. The reason why is semi-explicable; Oregon had two losses to Stanford's one. But here's why it's not: Oregon's extra loss came to LSU in the season's first game. So the Ducks were penalized for adding a more difficult game to their schedule, despite the fact that 1) Oregon won the Pac-12 and 2) beat Stanford head-to-head by 23 points, at Stanford no less. In the championship scenario, Oregon gets a place at the table instead of Stanford, which really didn't have much of a reason to be ahead of the Ducks in the first place.
Still, I think it's far easier to find negatives. In 2008, we're robbed of seeing Oklahoma-Alabama in one semifinal and Florida-Texas in the other in what surely would have been one of the more epic playoffs in recent years. In 2010, we get Wisconsin ahead of a dynamic Stanford squad whose only loss was to national title game participant Oregon.
It is worth noting that in two of the five years, 2007 and 2009, the playoff would have been the same in either scenario.