If the Ministry handled things more logically (like check Peter's and Sirius's wands for 'who cast the blasting spell'), and Sirius didn't go to Azkaban, he would've raised Harry.
Harry was going to know about Hogwarts earlier. He was going to know how his parents died, and he was going to be happy, cause he have someone like a parent, someone who understands him, someone who takes care of him.