I know practically nothing about Utah, but they probably have alfalfa in their crop rotation, because legumes bring nitrogen into the ground.
So, without growing alfalfa, they'd need much more artificial fertilizer, which can also have negative effects, like it being washed into lakes where it leads to excessive algae growth, ultimately leading to toxins being released and fish etc. suffocating.
I can imagine that there's some middleground which works better overall, but yeah, I don't think it's as simple as just not growing alfalfa...
Only legumes used in a cover crop MAY add nitrogen to the soil (temperature, inoculant, pH, etc has to be correct). If you harvest a crop from the field you are removing nitrogen.
For example alfalfa produces up to 500lbs of nitrogen per acre. It needs up to 600lbs/acre from full production. Alfalfa farmers for high production often add 10-20lbs of N after every cutting to make up for this.
The alfalfa is the main crop in those areas of Utah. The oats and barley are rotational crops. This is because alfalfa produces allopathic chemistry which inhibits the germination of its own seeds. So they plant oats or barley to allow the chemistry to break down in the soil for a year or two then back to alfalfa for 5-7 years.