I hope you're giving this woke lecture to the 98% of the fanbase up north that, under the crest of the Tribe, acts like total hating, dumb...donkeys. "I promise you" they do much more damage to the actual Ute tribe than any rival fanbase of a mascot by their name could ever do.
Right now, a "Ute" is primarily known as a jealous little-man diseased fan, uneducated but blinded by hate for others' religion and/or academic success. Yikes! Your lecturing efforts are much better spent there.
Plus nobody ever calls a member of that tribe a "yewt." We only call dumb yewt fans that. You should be happy it serves as a distinction and it should tickle your sensitivities every time someone distinguishes between the two by using the pejorative for the fan only.
Plus, regarding your last paragraph, I promise you they do say that a lot and none who use it think it's dumb