They do. Cats kill mice. And while domestic cats may have some difficulty killing rats, they definitely make life uncomfortable for rats. That's why the Egyptians decided 6000 years ago that cats are sacred.
Killing "small animals" such as mice and rats is a good thing. Ever heard about the Black Death? It was spread by rats (their fleas, to be precise). Yes, if we somehow managed to drive mice and rats to extinction, that would "reduce biodiversity", but it would be a good thing.