"Lynx preys largely on small to fairly large sized mammals and birds. Among the recorded prey items for the species are hares, rabbits, marmots, squirrels, dormice, other rodents, mustelids (such as martens), grouse, red foxes, wild boar, chamois, young moose, roe deer, red deer, reindeer and other ungulates. Where common, roe deer appear to be the preferred prey species for the lynx. Even where roe deer are quite uncommon, the deer are still quantitatively the favored prey species, though in summer smaller prey and occasional domestic sheep are eaten" Wikipedia
That's quite a lot of species that could do with control to set off against the occasional sheep. They seem to exist across large swathes of Europe without anyone being very bothered. Anything that would cut down on the population of rabbits, deer and squirrels seems worth thinking about