Problem is that in our area, there are very limited natural predators. The squirrel population has little control other than food supply. There are a few local birds of prey around, but not enough to keep the squirrel population in check.
Notes on squirrel population dynamics:
Squirrel population in the absence of predators is controlled by the squirrels themselves. They are territorial and fight for it. A big male especially red squirrels will control territory, young ones will sometimes share it for awhile, and females will drive off their young as a new brood comes in. You can act as predator but are really just disrupting territory and more will move in to the vacuum.