I am not sure that is correct in the case of wolves. In areas where they did become extinct coyotes seemed to take up their slack in the food chain. The thinking many years ago was that wolves were too aggressive with people and livestock and bounties were paid on them until they were extinct or near extinct in many areas- yet the eco-system did not collapse that I know of.
Where wolves have been extirpated, some species of ungulates have grown to unsustainable levels, decimating flora and also contributing to rapid spread of disease, like CWD and TB, which can and do infect domesticated livestock. Such diseases kill many orders of magnitude more domesticated herds than do carnivorous predators.
The North American Model of Conservation has as its cornerstone responsible hunting. It’s proven to have saved bison, raptors, ungulates, and many other species that are both critical for ecological balance and that hunters and wildlife viewers enjoy.
I think you’d find that most wildlife managers agree that domesticated herds need protection, but that exterminating entire species isn’t the smart way to do that.