For the sake of this argument, I'll assume that using the cloud works as intended. Here are a couple of concerns. The first is cost. Azure is a CPU farm but the demonstrations that use it seems to be better geared towards GPU usage. What this means is that Azure will have to throw a lot more horsepower, and thus money, at solutions than is necessary. Is it really going to be economical to use Azure for any game that wants it?
Second, Azure adds complexity to development. Kampfheld already mentioned that to do it right you have to code everything twice, once when your console has access to Azure, and the second time when it doesn't. This would continue Microsoft's transformation into Sony of 2006, where coding for the console is significantly more difficult than it's competitor. In addition, if a game always has to be able to fall back to offline mode, that means that using "The Cloud" can't affect gameplay in any meaningful way.
The only other option to requiring a split online/offline code base is to require a network connection for every game, even if it is single player. That is worse than the 24 hour check, and we all know how well that went over. It would inevitably cause problems during peak usage periods like during a game's release, because if enough capacity exists to handle that load at that time then it wold be wasted the 90% of the time when it is not needed.
In addition, places with poor internet connections could find this new breed of games unplayable. Imagine buying an XB1 because you love single/couch coop Halo and Gears of War, only to find out that you can't play them because the game requires a good internet connection even for these modes.