In many many places domestic cats aren’t much of an invasive species. In north and South America, Europe, Asia, and Africa there are similar cat species of similar size that take up a similar niche which were recently extinct (at least locally) many of which are closely enough related to housecats that they can interbreed.

One of the reasons to worry about domestic cats in some areas of the British isles is not to protect birds and the like, but to protect the closely related extant cat species that is getting its genetics replaced too much with housecat genetics. This wild cat used to have very broad ranges which disappeared.

Islands with no cat species don’t have this consideration of course.

Bird populations in places that should have and used to have catlike predators are and should be fine for the most part. Predators tend to help the population dynamics and health of prey species. Some people are just really uncomfortable with predation.