What if the music didn't start until you, the player, first attacked? Would that solve the problems with it? It seems as though just that would be easy to implement, along with an off switch to combat music.
It pretty much does. The Better Music System mod for Oblivion changes the battle music so that it won't alert you of enemies, and makes the transitions smoother too (not to mention dungeon type, location, day and night, and region specific playlists) and it makes it much better than the default system. It's also possible to switch any of them off.
Oh, and if you still play Fallout 3, try exploring downtown without background music. The fatigue sounds of the destroyed buildings are music enough.