I don’t think it even requires the police, in most US states, the law basically says that assuming the gated community has a “unauthorized vehicles will be towed at owners expense” sign somewhere, then the owners of the gated community don’t need police approval, they can call a towing company directly and have the cars removed, only needing to inform the police afterwards.
Unless OP is imagining a period in which police have much more pressing concerns than who’s piling cars up against a rich people neighbourhood.
I don’t think it even requires the police, in most US states, the law basically says that assuming the gated community has a “unauthorized vehicles will be towed at owners expense” sign somewhere, then the owners of the gated community don’t need police approval, they can call a towing company directly and have the cars removed, only needing to inform the police afterwards.