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.
Cattle were (usually still are) preferred among non-nomadic animal husbandry practitioners, precisely because of root destruction. Settled groups wanting animals which get to slaughter weight more quickly probably opted for hogs. Cattle rule because of their milk production, but hogs will eat pretty much anything, and not burn up a lot of energy because they tend to lay around a lot, just getting up to weight. Of course, they are by and large smarter when the day comes.