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).

Most rats are too big for the average cat to prey on. Mice sure. Rats get quite large.

Imagine trying to fight a medium sized dog. You'd probably win, but after 2 or 3 encounters like that you'd probably get hurt. That's a bad proposition when there's smaller, easier prey. And just like, human garbage full of food too.