Does the rivalry define BYU or Utah? No. BYU defines itself a lot by 1984, where a game vs. Utah was only a part. Utah defines itself largely by 2004 & 2008, where the BYU games were only a part.
I think people read the statement incorrectly.
As far as scheduling, my understanding is this: Utah has 9 conference games plus 1 B1G game per year. That leaves two games to be scheduled. If Utah wants to play Utah State sometimes as well as an FCS team (to help get 7 home games sometimes), they can't play BYU every year. It's as simple as that (and it doesn't even include adding a little variety to the schedule beyond USU/BYU sometimes, an FCS, and a B1G). With 8 conference games, I don't know that they'd say it was a problem. And in fact Whittingham said as much in an interview.