Most domesticated animals can adapt back to the wild, and do quite well.
Siro, it doesn't matter if rabbits or cats are cute. Liberals might be cute, too, but they can cause worse ecological disasters. It might be the law of unintended consequences, of plans gone wrong, whatever. Extinction of species is a part of the natural reality of our world.
We need a feral cat population to preserve the cat species. Keeping cats strictly "domestic" and out of contact with the real world, with heavy vet expenses and care, will produce a bad gene pool of cats selected by cat fanciers/breeders according to their tastes, or the demands of the pet supply markets.
The principles of genetics inform us that nature often supplies living things with a program for overpopulation, which is one way of efficiently selecting viable individuals for propagation. Nature lets hordes of marginal individuals starve, get eaten by predators, or die of disease, and reaps a stronger gene pool.
We take every little pet to the vet, and spend thousands of dollars on care, and capture and kill all the wild ones, and we get a rash of birth defects and a species that might go extinct in the wild.
Your favorite little bird, or the almost-extinct-already bird in the nature preserve, might be going extinct because no one culled the weak individuals for a few generations. You might actually save the birds by letting the cats have at them. Fewer birds will survive to propagate, but they'll be quicker and smarter.