The problem is, Fawkes fits a little too well into the Spaceballs maxim - "Evil will always prevail, because good is dumb". Fawkes certainly has a purity of intent that'd put any of the human characters to shame, but the consequences are not always quite so good as would be hoped.

(Incidentally, the comparison you drew makes me notice something - if Harry is searching for eternal life, there's a path to resurrection that neither MoR!Harry nor canon!Voldemort has noticed - phoenixes seem pretty good at that sort of thing. Mentioning them as an absolute contrast to dementors makes me wonder just how strong an antithesis they actually are, and if that might be an answer.)