I agree that the conference champion performing well in a bowl game does not help your favorite team immediately or directly, but a poor performance can reflect poorly on the conference. There have been years where the Big Ten has been something like 0-5 or 2-6 and bowl games and the social media will eat it up like they doing the SEC now for their bowl season start. I remember the 80s when the PAC 10 dominated the Big Ten in the Rose Bowl the same way the NFC dominated the the AFC in the Super Bowl in the mid-90s and it reflected poorly on the conference as a whole.
Still I understand the other side of the coin about not wanting to root for a hated rivalry, but even then, I've learned it is in the best interest for the Big Ten teams to perform well non-conference --- the direct benefits to RPI or KenPom rankings may not be as tangible as in college basketball, but I sense there is a benefit to the conference performing well overall in a way that benefits football poll rankings and access to the CFP or bowls in a given year or over time.