GreySorrel - You are mistaken about the basic effect of hoof type on range.
Cloven hooves will help aerate the soil. Domestic cattle and sheep, and even (truly) wild cloven hooved herbivores like pronghorn antelope, deer, elk, moose, mountain sheep and goats, all aid in the reseeding and regrowth of forage.
Solid hoofed animals like the horse, whether domestic or feral, compact the soil. This prevents new plants from taking root and, when it rains, the barren soil is more likely to erode.
Those who think the BLM is somehow picking on the feral horse herds are a bit nearsighted. Only looking at their favorite animal.
We need to appreciate that the BLM has a multitude of groups pressuring them. There are groups who favor the pronghorn antelope, the deer, the plover, several who lobby for the interests of various amphibians. Then there are the recreationists, and at the opposite end, those who want anything that wasn't here a millennia ago to be removed and no humans allowed on BLM ground. Including you and me.
Ranchers who do get a grazing lease are held to very strict guidelines. Guidelines that not only preserve the range for the grazing of livestock, but preserve and even improve the range for all wild and feral animals, and recreationists. The rancher, or his employees, are often the eyes and ears for the BLM and other groups tracking range conditions. Some of the other groups are governmental and some are academic.
It is much more complex than anyone picking on a species.