To the first part of your question: US lawns are not only monocultures, they are monocultures of non-native grasses. There's only a handful of species of grass used for lawns in the US, predominantly kentucky blue grass (native to europe, north africa, and west asia), fescue (native to europe), and bermuda grass (native to europe, africa, and oceania). none of these grasses typically grow as a monocrop in their native environs, they grow intermixed with other species of wild grasses, legumes, flowers, etc. North America has an abudant range of it's own native grasses, legumes, and flowers, but those aren't used for lawns. Legumes, like clover and peanuts, are nitrogen fixers, meaning they absorb nitrogen from the air and release it into the soil. grass needs nitrogen to survive, so growing any grass without legumes requires adding constant supplements of nitrogen fertilizer, and fertilizer usually has to be watered in. On top of that, because you don't have a functional ecosystem supplying nitrogen and consuming it in balance, and because you lack the necessary variety of plants for supporting other organisms, laws are essentially ecological dead zones where a sheet of grass is kept alive on constant life support. Active, healthy, richly biodiverse soil is great at holding onto water. The dead dirt under a monocrop is not. Because of this, you have to use way way way more water to keep your grass alive. Also, because none of these lawn grasses are native to the region they're being grown in, they are extremely vulnurable to inhospitable environmental conditions, with many of them being entirely unable to survive brief droughts. This means the only way to keep your *very expensive* lawn alive is to keep it watered in excess, constantly. Also, most lawns aren't even grown where they stand, they're bought from sod farms which deliver rolls of pre-grown grass in a soil matrix which you then roll out and water heavily and pray to god it takes root and also you should water it more to be safe and all the water is draining out between the sod and the soil oh shit you should really water it more, i think it needs more water.
To answer your second question: No, most of the US is not a desert, the deserts of the US are mostly limited to the west and southwest of the country. A quick glance over wikipedias list of the population of US cities indicates around 5-6 million people in total live in deserts in the US. That said, the definition of desert is variable, and you don't have to be in a desert to be in the red on your water budget. Los Angeles, the second largest city in the US with 3.9 million people, is not technically in a desert, but it doesn't have enough water for self-sufficiency. It can only exist because of an aqueduct that redirects a huge portion of a watershed from elsewhere in the state. Another river, the Colorado, doesn't even make it to the sea anymore because of how much of it is used. Even much more non-desert places can see issues with rainfall or snow pack that can lead to water shortages, and there are huge swaths of places in the US where groundwater is drawn up far faster than it can be naturally replaced (which is an exceedingly slow process) to the point that some parts of the US have been sinking to lower elevations due to the lack of water in their aquifers. And in ALL OF THESE PLACES, PEOPLE GROW FUCKING LAWNS. like. Lots of people do. Almost everyone in suburbia continues to grow and care for and overwater lawns. Even in deserts. Even in droughts. Even in places that see a handful of inches of rain per year. People keep growing lawns. It is the *default* that is expected of you.
If you want to know why it's the default, I'm not in a state to give that answer honestly, but look up HOA's, american suburbanization, white flight, levittowns, plantations, and french aristocracy. it's a long (and very racist) line.