Having had (pet) rats as well as a cat (briefly), I don’t think that cats are generally sufficient for rats. Fully grown adult rats are too big for cats to handle and they only take 10 weeks to reach a size that will make them troublesome for average cat.

Remember that a feral cat does not have access to health services. If they are injured trying to take down prey, they will probably be dead shortly after.

That having been said, if NYC has a rat problem, then it should also have a mouse problem. The problem here though is that trying to use feral cats to tame mice in the places where they’d be stubborn would be either troublesome, inhumane, or just disgusting (e.g. subway tracks).