This is a widely-cited myth, and almost impossible to measure in practice. What does "all the local wildlife" even mean? Is the threat here that birds are going to go extinct because of cats? Not likely, and the burden of proof is on the people repeating this mistaken belief.

Using "think of the birds" as a justification for imprisoning your cat for their entire lives is also pretty crummy. It's called wildlife because they exist in the wilderness. Even if cats kill a large number of birds, so what? Those birds don't have a happy, loving home with emotional bonds to an actual human.

If you think this logic is flawed, explain why you're fine with flies dying but not birds. I bet you've swatted a few in your time.