It depends.

My indoor-outdoor cat only catches small animals if they run between her paws. But she did chase a rather large raccoon around the house once, as I did.

In my suburban neighborhood, we occasionally have coyotes. They are known to prey on fat cats (the feline kind).

My feeling is that predation by domesticated outdoor cats is overblown.

I also feel that small wild cats were likely native everywhere. Birds were probably not their primary prey; small reptiles and mammals, i.e. animals that don't fly, nest in trees, or live in flocks.