The visit to the White House this week reminded me of a more direct involvement of a US president in selecting the national champion. The year was 1969. Going into the OSU-TTUN game there were four undefeated teams - OSU, PSU, Texas, and Arkansas.
On November 22, #1 OSU lost to TTUN to start the 10 Year War. This moved Texas up to #1 in the AP poll. After #3 PSU finished its regular season undefeated the next weekend, the players voted on which bowl bid to accept - Orange or Cotton. The players decided Miami would be more fun than Dallas despite going there the previous year and turned down a guaranteed game between #1 and #2.
Arkansas and Texas had their season ending game the next week with the two of them in the top two spots. The winner would be #1 and PSU would move up to #2 which is what happened. In those days, the Southwest Conference champion went to the Cotton Bowl.
Now for the presidential involvement. President Nixon attended the game in Little Rock. I don't recall if he sat on each side for one half. He came to the game having announced in advance that he would declare the winner as the national champion. He kept that promise as he presented a plaque to Texas after their 15-14 victory.
Needless to say, there were still bowl games to play and PSU was also undefeated. They were furious about the president dissing them. Texas beat Notre Dame in the Cotton and PSU beat Missouri in the Orange. The final rankings had Texas #1 and PSU #2 with both of them undefeated. The Buckeyes ended up #4 with USC getting #3 after beating TTUN in the Rose Bowl.