IANL but I think it’s slightly different than that. There are work visas available for seasonal agricultural workers, those who pick crops and such. I don’t know the visa period, maybe 6 months? They’re supposed to go home after the crops are picked and then re-apply. A lot of these are tied to a specific employer, though, so there’s the same problem as H1B visas - you can’t switch jobs easily. There are other short term visas that mostly go to people who work in vacation areas where there are labor shortages, like Cape Cod.
BUT the work of caring for horses or other livestock is a year round activity, so there aren’t visas that work for that situation.
So… There are a lot of people here illegally who do other jobs. I have read that the intense guarding of the border now is actually making people more likely to stay, because if you leave, there’s no easy way to come back. Before, workers - mostly young men - would come in, work for a while, save money for home, and leave. Now, they tend to bring their families and stay, because of the difficulty of moving back and forth across the border.