Would that even be beneficial? If the problem is that there are fewer insects and therefore the birds have less food, then cats suppressing the bird population might have a positive impact on insects, which the birds eat. That might be good for birds too in the long run.

Predators are part of a feedback loop. That the predators in question are housecats makes it a somewhat unnatural feedback loop, but that doesn't necessarily make it bad.

Housecats are part of the food chain as well; they can be killed by cayotes and raccoons, and hawks might sometimes go after a small cat.