My neighbor has an outdoor cat. This cat kills at least one bird a day according to their kids (the cat brings the corpse to their porch). So I think you're tremendously mistaken about "not at any great rate."

Housecats are not native to most countries, and feral cat colonies are not healthy either (lots of disease). They are not part of "natures balance" and we're not eradicating them to extinction. That's just utter nonsense.