Birds that get eaten tend to have high reproductive rates, and if predator levels drop their population gets so high they start to die from hunger or disease (which spreads faster in crowded conditions). Many predators eat an incredible number of animals - a large owl may eat several small birds every day for itself, and a multiple of that if feeding nestlings. Cats aren't unique here.

Domestic cats can be a threat to specific populations already endangered by other factors, but usually those are ground animals. Cats being a threat to population levels of birds in most of the world just isn't believable to me.

I have always suspected it's a minor bit of public misdirection to keep the stray cat population down in North America. Other parts of the world have much larger numbers of outdoor cats and their bird populations are no worse off.

Domestic cats that go out to hunt aren't controlled by natural causes, like food availability for example. But they go out at night to hunt nonetheless, and they're usually in their peak conditions because they don't have to go hungry for long periods of time.

As numbers that I am aware of,

Cats kill an estimated 50-200 million birds annually in Germany, according to NABU and Max Planck Institute estimates

With a cat population of Germany approximately 15 million, most of which are permanently indoor, the rest are keeping themselves very busy for no reason