The full extent of Harry's abuse at the hands of the Dursleys comes out when Professor McGonagall hand delivers his Hogwarts letter so, by the time Sirius gets out of Azkaban, Harry is living with the Weasleys and neither needs nor wants a godfather. With nothing left to hold him in the wizarding world Sirius accepts the chance to try and rebuild his life again in the muggle world.

Harry is deeply depressed after the Department of Mysteries fiasco and doesn't know what to do with himself. Then, though Fawkes offers Harry the chance to go exactly one year back in time, doing his OWL year all over again, saving Sirius and better preparing himself for the battles that lie ahead. Obviously Harry jumps at the chance.