The reason those predator species got killed to near extinction is probably a bit more broad and ugly than just concern for lifestock.
But you’re correct of course that the main reason for the current overpopulation of many non-invasive species is a lack of predators and they should be reintroduced. But that’s also not a thing that can happen from one day to the next, so even in the most optimistic scenarios some degree of human wildlife management still has a purpose.