I think that the mechanism for turning the LCD off when the eDGe is closed operates when the hinges bend in a certain way. If you power up while it's shut and it remains shut, then the eDGe doesn't know that it's shut.

I still don't understand why you would turn it on and leave it shut for long enough to completely drain the battery... as previous posters have suggested, why not simply not power it up until you're ready to use it?