The tragedy of Achilles isn't that he got mad and was punished for it through the death of his friend.

The tragedy of Achilles is that it took him ten years and a very stupid argument to realize that dying in a pointless war against people who never wronged him was a complete waste of his life and that he valued living over glory, but at that point it was too late to disentangle himself from the cycle of violence that would claim his best friend/lover and inevitably himself.