The only games were you "do" something to reveal the story are the ones like Heavy Rain or LA Noire or the visual novels where the characters actions involve things like making dialogue and plot choices.
The other 99% of games' characters abilities are limited to things like: "shoot this or don't shoot it," "jump over this or run around it." Kind of silly to make meaningful plot developments EMERGE from THOSE activities.
I am also of the opinion that games with ONE plot tell the best stories. If that means relinquishing control of said plot - I'm cool with that. Let us, the gamers, be immersed and have control of HOW the character advances through the plot, not WHAT that plot is.
... again, just my perspective and you are entitled to yours. If you can explain how yours can and should apply to (most? all?) games, I'm all ears!
You say so yourself that LA Noire and Heavy Rain aren't games (visual novels--might as well be choose your own adventure books) so, without a framework of rules and objectives, I don't think the menial tasks the games make you perform count for anything, all they do reveal more non-interactive scenarios.
You seem to think that the framework of a game should serve the plot. But this is totally backwards. Is LA Noire a chore to play? YES. Same goes for every PLOT driven game I've played. Killer 7 is another good example. The problem with these games is that they don't want to be games and they take this problem out on the player. A game has to logically respond to player input-these games simply shrug off all wrong input until the predetermined right input is done--it's not interaction of 99.99% of the actions garner no meaningful reaction. The goal of these games basically "do what you're told." You can't build a strategy around this because your role in the game isn't big enough to effect the game.
Think about the Blue Shell in Mario Kart and why someone might hate it. The blue shell punishes good play and rewards bad play. There isn't a logical strategy to overcome the blue shell. The skills of player are diminished.
Stories that emerge from game play are generally more personal and effecting. Minecraft is a good example. The fear night brings. The satisfaction of building a cool house or finding cool stuff to mine. Being annoyed at your dog--being sad when your dog dies. The framework provides so many little moments that build the emotional arc of the game and thus the story is created. Sports games (especially full seasons) do a good job of this--same goes for the Sims or SimCity2k and the like. I've heard Dwarf Fortress really excels at this.
Games are a great medium for telling stories but the games and the stories tend to suffer when the games ape movies. Adding movies to games hasn't made the framework or game play of any game I've ever played better. There are plenty of effecting non-plot driven works of art that are deeply moving--statues, architecture, paintings, songs, poems, pop-art.. etc--given the spacial reasoning and non-linearity that most of these art forms possess, I don't know why developers insist on making pseudo movies that cripple their chosen medium--games.