I'm the first to admit that my perspective is a bit different since I grew up a Utah fan. Not something I try to hide. It's strange to me that having been a Utah fan when I was 15 though disqualifies me from ever becoming a BYU fan when I attended there.
I also have a pretty national perspective (Air Force, currently living in Georgia) where I see that neither BYU nor Utah are a big deal or topic of conversation among fans. That's not a Ute take (I'm on some of those boards as well and they all think they're more relevant than they are as well), that's an honest take.
BYU has been seen as effectively a G5, so it doesn't matter how well they do, they're not relevant to most fans as anything more than an occasional feel-good story. Utah is a P5, but not one of the ones that really matters. Minnesota, Kansas St, Pitt etc. They're all fine teams and can occasionally win big games, but nobody outside the fanbase or the conference follows their weekly schedule.
BYU now has the chance to hopefully change that narrative (their part, not Utah's) here and I actually think the Big 12 is a great shot to do so. It's a conference without any clear top dog (BYU, Cincy, UCF, and Oklahoma St are probably the 4 best) so if BYU can start winning conference championships in that conference, there's a chance that we can become nationally relevant beyond our conference.