The cat population can be simply and easily controlled with the introduction of some eagles...
You're right about the toll on the birds and other small animals, but not about the exploding population.
Key West has an informal program of spay/neuter of feral cats. They also inoculate them and notch their ears. It stabilizes the population.
Fast forward 5 years, you got yourself a healthy plague of feral cats - plus complaints from regular cat owners that the ferals pick fights with their cats. In addition introducing mass numbers of cats causes a rapid uptick in bird deaths (cats can eat islands virtually birdfree) and they're a considerable risk for accidents with vehicles or small kids.