Harry was being kept at the Dursley's for his own protection. Chapter 37, Order of the Phoenix.
‘While you can still call home the place where your mother’s blood dwells, there you cannot be touched or harmed by Voldemort. He shed her blood, but it lives on in you and her sister. Her blood became your refuge. You need return there only once a year, but as long as you can still call it home, whilst you are there he cannot hurt you. Your aunt knows this. I explained what I had done in the letter I left, with you, on her doorstep. She knows that allowing you houseroom may well have kept you alive for the past fifteen years.’
Thus, if Harry suggested he live elsewhere, I'm certain that would be quickly quashed by Dumbledore, just as Harry's plan to live with Sirius would have been. Why did Harry never attempt it? As a general rule, even in bad circumstances, Harry rarely shows initiative to change them.
- He never challenges Snape's treatment of him, to, say, Dumbledore, despite it being unconscionable.
- He never requests help from anyone in proving that he wasn't responsible for the basilisk attacks, nor that he isn't the Heir of Slytherin.
- He never appeals for help with his Dementor issues during PoA, except to Lupin.
- During the course of the Triwizard tournament, he never asks for special treatment because of his age, nor does he seek a way out of competing, by, say, not participating in any of the Tasks.
And so on. In the same book when he begins to show initiative, book five, Dumbledore explains to him why he has to stay with the Dursleys, so Harry does.