In terms of weed control, I think your best bet is spraying. It’s just a practical reality of maintaining pastures. No, I don’t love it, but it will take care of your problem.
Keeping sheep is like keeping any other type of livestock. You need to learn about them and be prepared to care for them. The hardiness of sheep depends widely by breed of sheep, location, and local predators. Some breeds of sheep are close grazers while other breeds are more “browsers.” Browsers will definitely eat down the rough areas where your horses poop and pee and will eat quite a few types of weeds (not the toxic ones).
Sheep require a more secure perimeter fencing than horses, but are not typically escape artists. Re-doing or supplementing your fencing might prove an expensive investment. Sheep do require protection from predators. That generally means a livestock guardian dog. That’s a whole other animal that needs to be raised, trained and cared for (and contained–without proper training many will be prone to roam).
It’s difficult to find a vet that will work on sheep, so you need be prepared to do your own vet work. Obviously they need vaccinations, deworming, and periodic foot care like any other hooved animal. It’s not always so easy to pen up and restrain a bunch of sheep, with or without a chute setup–depends on your sheep.
Whatever the books say, many hair sheep do in fact need to be sheared. Other people’s opinions may vary, but IMO sheep shearing is a pain.
I have never known a horse to be afraid of sheep, but some horses will kick sheep which can be deadly for them.
All things considered, spraying would be a lot less expense and headache.