Having different teams work on the same world isn't a strange thing in any kind of writing. It comes down to the type of game and the type of story more than anything. You can put pretty much any kind of game and any kind of story in the ME world, and if you have the right people(from writers to programmers) for that kind of game and story it'll probably come together very well. That's what ME is. It's the world more than the game.
People have voiced some crazy wariness, and probably rightly so, about using any characters from the game, but this is only a thought exercise. Picture Garrus, but not in a grand epic space opera game. Picture Garrus in a Max Payne style story where he's trolling the criminal underworld. Picture Garrus uncovering the mafia or the smugglers or the god knows what else who were involved in a murder case he could never quite solve when he was C-Sec. Now you either have his journey to becoming Archangel between ME1 and ME2, or you have his return to being Archangel after ME3. And not in a grand epic way. Just a personal story about Garrus where he goes gunning for the bad guy.