I don't rush the process. Puppies don't really have much in the way of impulse control. Something runs by and they chase, it's what many have been bred to do, chase the rapidly retreating object. I let my pups drag a leash when loose so I can stop chasing and I also set up a crate or ex-pen in the living room so cats can be a round the puppy and still be safe, and the puppy can get use to seeing the cats around.
I don't expect my cats to "train" the puppy. Training the puppy is my job not theirs. If it was a child I doubt anyone would recommend just letting the child teach the puppy not to chase and bite it, that would be the job or the parents not the child.
If all you have is a hammer, everything looks like a nail.