As of "now", the Big12 still carries the P5 moniker, which the AAC doesn't have and the Big12 will hold on to it through the end of the current CFP contract. What happens after 2024? Who knows. But, for now, the Big12 has more clout than the AAC. Further, the Big12 has only one "dog" program in football, Kansas. The AAC has several ... ECU, USF, Temple, Tulane, Tulsa.
Further, if the CFP proposal is modified from the current "six best conference champions" to the "five best conference champions", it behooves the Big12 to take the best teams from the AAC and MWC to cut them off at the knees, even if it means going to 16 teams. So, take Cincy, Houston, and UCF from the AAC; BSU and SDSU from the MWC; and BYU. That turns the Big12 into a 14-team conference and effectively kills both the AAC and the MWC in one stroke. Further, this "new" Big12 would compete for the #4 spot versus the PAC in most years.