Not necessarily, if you're implying that cats predate birds efficiently.

Cats are fairly lax, in a manner of speaking, in catching birds. Mice, voles, gophers, and rabbits are easier to catch and more plentiful. We'll see them catch maybe one bird a month.

In contrast, the number of birds in our area is innumerable. We'll get hundreds at our feeders every day, and see dozens of songbirds (Robins, Atarlings, Red Winged Blackbirds, Magpies, Woodpeckers, Pigeons, etc.) while driving into town.

And that's without getting in the raptors (some of whom are a threat to cats).

I imagine humanity's reduction of bird's habitats has a bigger impact than the carnivores we bring with us.