I have a feeling this is going to vary very strongly between areas, and between cats.
We have a cat who only takes the occasional rabbit, and nothing more. She even leaves the several pairs of assorted water fowl nesting 20 yards away alone. She is happy and healthy at least 15 years old as a cat who spends most of her time outside.
And for Europe, cats are native. The European wild cat has been driven virtually to extinction by interbreeding with domestic cats. I don't know about the US.