It’s a standard line among human hunters: a good hunter eats what he kills—which means that you don’t kill for the sheer pleasure of killing, leaving a dead beast to lie and rot. Well, a cat in the wild definitely follows that rule, devoting its predator skills to finding food, not engaging in sport.
But well-fed house pets don’t follow that rule at all. They gladly kill when they aren’t particularly hungry, and they will even kill nasty-tasting things like shrews, which they have no intention of eating (and which, if the cat was hungry, he wouldn’t bother pursuing).