Agreed. I think somewhere along the line, people swapped "remain calm" for "don't react".
You see, a dramatic reaction is bad. A raised voice or a shriek will potentially encourage your bird to bite. Keep in mind that a bird's method of communication differs largely from our own. "Ow!" in a raised voice is a no-brainer for us, but is not necessarily recognizable as a bad thing in the avian world. You have to put aside all assumptions that your bird will understand things in the same way we do.
So if you were to scream or shout when bitten, your bird might find it amusing or curious. Either way, he'll want to employ the scientific method to test his hypothesis: bite=funny sound. From this, everyone figured, "Ah! So we shouldn't react!"
Wrong.
Watch any nature show, ever. When was the last time you saw any bird just stoically take an abusive behavior? Exactly never.
No, what you do is remove the beak from your flesh while saying, "No," in a firm, yet even, tone. And then you immediately put your bird on timeout and ignore him COMPLETELY for the next few minutes. And I mean completely. Back turned. No eye contact.
After maybe 10 or so minutes, you approach him again. At the first sign of aggression, right back on timeout. But the instant his behavior becomes more peaceful make sure to reward right away with a treat. If you're consistent with this, he'll begin to associate biting with being ignored and alone... and good behavior with treats and positive social interaction. (This is where your point about parrots being social creatures comes into play.)
Make sure to check out those links that Wendy provided. There's good stuff in there.