I can understand the damage inflicted by cats in environments that never had a very efficient bird predator like cats (e.g. New Zealand and other islands). But in North America there's the Bobcat, Lynx, Ocelot, Margay, and Jaguarundi. Even without the domestic cat, birds would still get eaten in sizable numbers by cats. Sure there are a lot more domestic cats out there, but without civilization there would be a lot more wildcats out there eating birds.
Maybe the impact is simply that outside house cats exist in numbers that are not supportable for a normal wild predator due to the fact that house cats have meals provided by humans to sustain them. So house cat numbers don't reach an equilibrium with their prey the way a wild cat could. Either way I suspect that the impact on bird populations by domestic cats is often overstated.