Before they bolted, the TexAggies were in an uproar over the statement that Longhorns see the Sooners as our biggest rival. Nobody said A&M wasn't a rival, just not the most important. Instant identity crisis on TexAgs. They listed all the stuff we do regarding them (their line in our song, hex rally...okay, the two things we do) as proof.
So, we start talking about it on whatever the principal horn site was at the time. My thought was that it's a bigger deal to beat OU and a bigger deal to lose to A&M. That's pretty much for the reasons you state.
Relatedly, when Nebraska rose to prominence, OU's game against them was huge every season. The games were great. I wondered if Sooners considered Nebraska their main rival. I honestly didn't know their history. You know what? I didn't really care. Texas-OU was still one of the great games nationally and I've never been present at any game that meets the intensity of that game being played on a hot day.