So I tested this with just two games(I have a 40'' tv), first one was Indigo Prophecy which supported fullscreen 1080p from start. After that I popped in Star Wars Episode 3, that game did not support it. So I had a look around on the internet, and found that you have to change your display to 480p. So I did, menu looked awful after change, but the game then ran in fullscreen.
Just change your display back when playing an actual Xbox 360 game or doing something else like netflix