Yeap........... You are right they will and do eat anything that won't eat them............. And there lies the issue they have no predators except man, Kepp letting them breed and one day soon you will wonder why you have NO small game and the deer just don't seem to be as numerous as they once was.............you are talking about a predator that is not native to the area......... atleast not for many, many years.
Count up all the predators you have in your area..........it's likely to be something like birds of prey, coyotes, bobcats, and apparently mountain lions............
But it's your area.......... who am I to care if you would rather have large predators on you land and less game animals...............
Three can keep a secret................If two of them are dead! (Benjamin Franklin)