You can't make someone root for a team he just can't root for.
But, as a matter of fact, when the conference champ does well in a high-profile bowl game, the glow rubs off on the whole conference.
That worked for the Big 12 when OU won the 2000 championship and when Texas won the 2005 championship.
It worked for the Mountain West when TCU beat Wisconsin in the 2010 (season) Rose Bowl.
It worked for the B1G when Ohio State beat first Alabama and then Oregon to win the 2014 championship.
It has worked for the SEC as Bama has won 4 of the last 8 championships. Hence, two SEC teams in this year's playoff.
And that's not due to the fans or administration or media of the victorious institutions generously passing around praise for the conference or the other members. It's just how it is.