They certainly do, if given the chance. Boom-bust cycles caused by resource overextraction are common throughout the animal kingdom. The only reason that it isn't apparent to you is because you don't pay a lot of attention to animal resource supplies, and because over a long enough timeframe new equilibriums are sometimes formed until some other external impact comes into play.
Re:hunting for sport, cats are the most famous example as mentioned by the sibling commenter, but you also see it with other predators. Play is very common in the animal kingdom and its often violent. I've seen coyotes and foxes kill smaller animals and leave them uneaten many times.
Humans are simply better at resource extraction than other animals. The underlying behavior is not notable.