…except the local bird population.
Wild predators don't consume more than the environment can sustain
Sometimes they do
Wild animals aren't less taxing on the environment per se, they're just differently taxing. Long living ecosystems are likely to be closer to balance. But never perfectly balanced.
Domestic cats are just cats same as the wild ones, but in different numbers and locations
For example, people like to talk about beavers as good for the environment. Sometimes they are, sometimes they aren't!