I too disagree. With all due respect, I deeply understand the idea of preserving "the brand", but in reality the entrenched opinion among the rest of the nation (and world in my experience) is it's all one BYU, just like the State Colleges of New York or California systems which are massively larger than all BYUs combined.
I've never met anyone in all the states I've lived in that differentiates one BYU from another geographically. The second the acronym "BYU" (of any kind) is mentioned as a frame of reference, you're immediately "pinned" a member of our Church, with all the good that comes with it. Outside BYU circles, people don't split hairs over the academic prowess , rigor or focus one one or the other (though their purposes are notably different). One thing is for sure...The kids at BYU-Idaho have HUGE watch parties for all sports, wear all the BYU swag, and more. It's like their own ROC... super, super fun.
If anything, my experience is that BYU-I kids are perceived as more well-rounded, disciplined, focused, self-reliant, less "pin heady" or entitled. They're 80% there for the quality/value of training and 20% for "the brand"...and not the other way around. BYU-Provo, BYU-Salt Lake Center, BYU-Hawaii, BYU-Tdaho, Ensign College and Pathways are all incredible and serve different purposes that well and all constitute the true "brand" of BYU.