I think the 'cloud' thing Microsoft is talking about has nothing to do with Gaikai. Please correct me if I'm wrong, but in no moment since the reveal they mentioned full-blown streaming of games being run remotely. At most they commented on the possibility of P2P streaming for allowing a friend to take over the game remotely in order to help clear a specific section.
MS mentioned that "every" game gets dedicated servers and that the XBone uses Azure (which doesn't have GPUs, AFAIK). So instead of having one of the players host a P2P match, they can spawn a server-only build of the game on a virtual Azure server to host the match, allowing developers to completely avoid annoying problems inherent to P2P like host migration, host advantage and host overhead.
Since the server doesn't need to do any graphics, it can uses far less RAM than the client build and several of them can be ran on the same virtual machine without problems. 300k virtual machines can easily handle 6+ million concurrent multiplayer games.
Why does everyone seem to think streaming video of a game from the cloud is possible and makes a playable game, but streaming other data is impossible?
Are the 1s ans 0s different?
Streaming other data is possible. Streaming data vital for rendering graphics of the current frame locally and other part remotely is not: the client will complete their job in a dozen or so milliseconds while the server will take 30+ milliseconds to even know there is work to be done. What will the client do? Sit and wait? Start rendering part of the next frames? But the player didn't saw the result yet, what input data is the server going to use? If the client is having to delay it's output while it waits for the server output, why not do the whole thing on the server anyway?
The most a remote server can help with is with simulation. For example, an open-world game where each individual NPC has a very detailed dynamic simulation of their daily routine. The client can only see a fraction of the NPCs at a given time, so the simulation for all NPCs could be done on a server, like a MMO full of bots instead of actual players. However, it's a complete waste of money to spend a dedicated server for a single-player game and such dynamic persistent world would be much better used in a multiplayer setting.