I , do/do not agree about Arizona grass lands, if I understand what is being said.
Our deserts as they are today is do to man's impact. Read 1493 by Charles C. Mann and his other books on the subject. We speaks largely of the eastern part of this continent. The Spaniards did their part from the moment they landed. Subject being land, they brought with then in settling Texas long horn. They came in the first place because of the long high endless grassland, stretching north, east, and west without end. It reminded them of Spain. Wealth was cattle, not in coin, more cattle = more wealth. However, it is not the easy to get your cattle to Spain in 1500's -18's. fencing was not nessesary so they muliplyed and eat. The land was grass because there was balance. (Not going into that). Over grassing, slowly turned the land dryer then what is already was. Then, being lonely for home they imported plants for their gardens. Tumble weeds was among them. Seeds blow away and self planted, took the little water there was from the grass, no grass, no cattle, then no Spaniards. Only a handful remaind, by the time people from East moved West. Those that moved West fishihed off the land in a few years to what we see now. Tucson is very interesting. Most of that area was a forest of desrt trees such as Misquite. We cleared the forest in short time, for mining and building the towns. Being a desert tree they grow much slower then other hard woods. There was no chance of it surviving. The wast high grass that covered our Arizona lowlands, quickly grassed away. Seads in larger quantity hitchhiked hear much faster then the wind.
Ofcourse, in this extremely short Synopsis, I skipped 99% of the story.
We have always (at lest for a 1,000) years been classified as a desert (I think that is 9" of rain average). Easy to through off balance.
Many country's do use sheep for cutting grass. The military and our goverment use goats in sensitive areas to maintain a low fire risk. I think it would be good for us to be shocked today by seeing sheep cutting grass on public land. Even better if we eat more mutton and lamb, less corn feed beef. But I'm only one.
PS: Oklahoma still practices the Native Tradition of burning the prairies in taking care of the land.