Except if the animal is quick sometimes they get a paw caught in the trap instead of their head, and you then have to deal with a crying, injured little furry rodent . ..
I've had success with getting rid of moles in my yard just by flooding the tunnels every day with the hose until they decide it's too wet where they are nesting, and move on. I suppose I might have drowned a few little critters, but for the most part, they just migrate next door. Then my neighbor comes over and says "what do you do about the moles in your lawn?" to which I innocently reply "What moles?".