It most likely has to do with copyright issues. People could just stream TV shows and movies as well. It wouldn't really matter though, because playing 360 through HDMI in is completely awful and totally unplayable for pretty much every genre of game. --- Demon to Some. Angel to Others.
Why is playing through HDMI awful and unplayable? I have no problems with it and the display looks great.
Because unless you're playing something that doesn't require twitch reflexes (or any sort of reflexes), it'll be hampered by the input delay along with the extra input delay added by your TV if it doesn't have a Game Mode display option. --- (>O_O)> <(o_o<) (>O_O)> <(o_o<)