The first thing to bear in mind is that the error message "The disk is unreadable" does not necessarily mean what you think it means. It is a "catch all" read error that is displayed when the Xbox's operating system cannot read data from a device. That device could be the DVD, the HDD, or a USB. The problem could occur when Skyrim is reading game data itself (i.e. loading a level or software), a savegame, or a file in the game cache.
So, when you say "the disk is unreadable on an installed version", the game (or the Xbox operating system) could well be having a problem with the HDD.
Because the error message is so vague, and each players console and setup is different, it is difficult to know (given the limited information provided) exactly what is causing your particular problem. However, here are a few things to consider;
1) When exactly does the error message appear? Most likely, it is happening because of a corrupted save file (i.e. the error message pops up after loading Skyrim and choosing a device. At this point, the game will attempt to collect information on savegames, and if your latest savegame is corrupt, it will display an error without it actually telling you it is trying to read the savegames. Hence, you may think it is the disk that is going wrong). If this is so, try moving your latest (only) savegame onto a different device because it could be corrupted (do this from the dashboard). Then, try loading the game again. If the error occurs again, repeat the same procedure. Keep doing this until the game does not display the error (assuming your savegames are corrupted).
2) Savegame corruption can (and has happened) at any time (i've had it myself). This may or may not necessarily be Skyrim itself that is causing the corruption (there is a small chance that your HDD is wearing out!). I would assume the problem is with the savegames, and not with a problem reading Skyrim or the DLC itself, because you mentioned that when you deleted all of your savegames, you could play the game again.
3) In general, If you are writing your savegames to a USB, I would avoid that because of the possibility of a "read disturb" with USB devices. What this means is that data can be corrupted simply by a USB access to nearby data. I did used to have savegame corruption when I stored my games on a USB, but I have not had a savegame corrupt since keeping my saves on the HDD.