The population of Utah is up to 75% LDS, depending on the region, and the LDS church advocates taking part in the political process (unlike many evangelical churches, who believe that the Bible has no guidelines for political action or causes). Given that the US has a representative government, why should Utah elect anyone who wasn't LDS?
I have lived in Utah briefly. I didn't particulalrly like some of the Utah culture there. (There are a lot of things I liked) There is quite a bit in Utah culture that is not part of the LDS Church. But your over generalization was innacurate and based on what? Since you have never lived there.
There are plenty of non mormon politicians. Unfortunately most are Democrats as well as some "Mormon Democrats" which I can't understand. Look here just go to utdemocrats.org (Don't want to link it to promote the Democrats in Utah but here it is for the curious.)
The town I live in currently is predominantly Catholic. If I looked hard enough I could come up with some wild conspiracy theory about how the "Catholics" run the town since most of the town council, business owners, police judges etc. are Catholic. If a person cuts me off while driving perhaps I should blame the Catholic church? But yet here I am a religious minority and I can still vote and enjoy my constitutional liberties. There is no Catholic "Cabal" oprating in my town. But yes it would probably affect the local politics based on Demographics. That is why I sepcifically asked if you had a personal experience where your liberty was infringed upon. Apparently you do not.
Most Mormons are Conservative Republicans or Constituionalists. It is the most Republican voting state in the union. Here are two counties I found from wiki (too lazy to link).
This county is often referred to as "the most Republican county in the most Republican state in the United States." In the 2004 presidential election, 85.99% voted for George W. Bush. [1] However, this title is inaccurate as Rich County, Utah cast 88.91% of its vote for Bush.