Horsebased societies had to constantly be on the move to find good pasture for their horses. Cattlebased societies don’t have to move because cattle can suck any nutrient out of just about any fodder or silage as cattle have to gut to do that. Coupled with metallurgy, it was a weapon of mass destruction. End of story.
Goats and sheep are probably the best animals for sparse vegetation as feed. They eat down to the roots. And they don’t need as much feed to live on since they are smaller animals.
Cattle and horses will starve to death where goats and sheep can live.