The vast majority of the American West cannot support agriculture without irrigation - often involving water from wells. It is a dry place, and this reality sets it apart from the other North American regions. It is depicted as a dry place because that is what it is. There are obvious exceptions to this, notably the Pacific Northwest, the coast in general, and various places hugging mountains where precipitation can be at its greatest. That said, crossing the expanse - by wagon or now by car or train - involved/involves a great deal of time crossing very brown terrain. Because water is key to settlement, the result is that much of the Intermountain West has been characterized as a sagebrush ocean - vast amounts of unsettled land with isolated "islands" of towns. It also has resulted in Nevada, for example, frequently appearing as the second-most urban state in the nation (the first being New Jersey - which has nothing to do with this modeling!).

Surviving journeys could be extremely difficult and resulted in many deaths along the California-Oregon trail. Crossing what was called the "Great American Desert" was not easy. People followed rivers as much as they could, but in Nevada especially, there was a forty mile stretch with no water, tackled at a point when the oxen were at their weakest. If you go back to the 1850s in a time machine, stay on the East Coast, or arrange for your teleportation so that you arrive in San Francisco. San Francisco has always been a great place.

In addition to the other great answers here, it should be pointed out that part of the "romance" of the American West in the 1800s - even in contemporary books and newspapers - was that it was distant, wild, foreign and exotic. Deserts embodied that "western" environment very well because they were very different from the landscapes of the Midwest, South, or Northeast. No place in New York looks anything like Monument Valley. Until a few generations ago, the odds of someone in Ohio ever having seen the Sonora Desert were very low. It might as well have been on the moon.

The non-desert landscapes of the West, on the other hand, were too familiar-looking to people who lived back east. A forest in Colorado can superficially look similar to one in Pennsylvania. A beach in California might look like a beach in Florida. It's easier for us to distinguish between these places, but decades ago people weren't that savvy about such things. Those landscapes are similar enough that many California locales are still used as film and TV stand-ins for other places in the US and around the world.

So from a pop culture standpoint, when you wanted to portray a place as "western", showing a desert was the quickest and easiest way to do it, even though much of the west doesn't look like that. And deserts come loaded with their own sorts of drama too: lack of water, barren vistas, little shade or shelter. Deserts provided a built-in level of exotic danger for easterners that familiar places don't. And once the desert became the iconic location for "western", it became what people expected to see, hear, and read about, so it eventually became a self-perpetuating cliche.