Either there are quite a few self-loathing LDS Ute fans, or EVERY SINGLE non-LDS Utah fan hates the LDS church.

I think you are wrong. I think it is about 5% that truly "hate" the church. There is another 10-20% who don't hate the church but certainly roll their eyes a bit and/or make jokes that some might find offensive. They are a vocal minority, but a minority none-the-less.

Some of the jokes you point out are offensive. The sacrament cup jello thing was in very poor taste. But I also think that we as LDS people can stand to have a little bit better sense of humor. This is one thing I truly admire about the Jews. They poke more fun at themselves than anyone else.

I've seen signs at the Huntsman Center that say, "The prophet is a ute." or "Family Home Evening=Losing to Utah" (for a Monday night basketball game back in the day). Is that really offensive? We aren't talking about jokes about Jesus or sacred things like the sacrament. I think those are par for the course in a rivalry.