@Anomander the problem is that account data is tied to the device. Sure the data is stored on the cloud server, but if the terminal can't tell it's you, then it's useless. Now if you had paired it with some form of account or social media it would be fine.
To make an analogy, let's say the game is a house. All the data is furniture inside, and when you play you open the door and go inside to, I dunno, sit on the furniture or something. Now the door locks to make sure that you're you, and you sign in or unlock the door by telling the server who you are. If the key, in this case, is your phone, then you're screwed, because the key is broken.