I wonder to what extent domestic casts have replaced mustelids (weasels, badgers, etc.) and similar small predators in North America. If there were fewer cats, would there be more mustelids? Would they then have a similar effect on the bird and rodent populations?
I imagine they make a poor replacement for mustelids. Mustelids tend to prefer small mammals and birds, but they usually vary their diet with insects, fruits, some plants, carrion (I think cats generally don't go in for scavenging, not sure) etc. They also have different hunting approaches: weasels will go into burrows or dig into vole dens. Cats wouldn't be able to fit, and I've never seen a cat actually dig to get at prey. So the small mammal and bird population is adapted to predators with different tactics. Plus cats have a habit of hunting even when not hungry, which amplifies their effect on the ecosystem.