I can tell you for free the numbers of birds that cats get are a fraction of the numbers of things that don't fly. And the numbers of birds that the average cat gets that are not of it's own kill are higher than you'd think. They might be natural predators but your average domestic is also as lazy as F and won't put in the effort required to seriously dent the bird population. Having said that you do get outliers like with any statistics, but locally the domestic cat population (with the appropriate motivation) does more to keep the rats in check than just about everything else that humans are doing - this is in a lifestyle block area which is one of the worst culprit landholding types for reservoir populations of rodents etc.
Rats do more damage to bird populations than cats - as they tackle the eggs which removes all chance for the breeding population to breed and also once the pairs are stressed they tend to move out of the area which creates more issues.
On the other hand, feral cats are utter bastards and there is no place for them on this god's green earth.
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