I don't think his point is that it isn't cathartic; he's pointing out that there is no physiological reason for crying in particular to have evolved to be cathartic. What have leaking eyes or a quavering respiratory system got to do with relieving overwhelming emotions? Nothing. And no other animals experience this.

It's a confusion of cause and effect (or post-hoc fallacy). "I cry because it makes me feel better" supposes that feeling better is the purpose of crying. But really it's the other way around: Crying is the purpose of feeling better. Crying is the thing with purpose, because it's an important and useful social signal. Overpowering urges to succumb to it, and feeling better after having done so, are evolution's way of coaxing our minds to do something that's useful to us, just like urges to eat or have sex, and feeling better after having done them.