As cofree suggested, I also think it sounds suspicously like a laser problem. DVD's are "easier" to read than games, as are original games easier to read than backups.
So if it is reading DVD's fine and your original games I would guess at the laser.
Restore the drive and try an original game in the drive. If it plays that then we know the drive is good, but it is probably just the laser giving you problems reading backup media.
If this is the case you have two options:
1.) There is a good tutorial showing how to adjust the Pot of your drives laser. ie: Calibrate the laser to increase its power/strength to help it read discs. You will need a multimeter to do it. If you have one or can get hold of one, I guess you have nothing to lose in giving it a go. As long as your careful & don't damage anything when taking the drive apart you can always put the drive back as before if it doesn't work. Tutorial here:
Xbox360 DVD Pot Calibration
2.) Dump the firmware from your drive, make sure you have a good dump by checking the Key in Toolbox 4.0. Then buy a replacement drive off ebay or similar & flash it with your firmware to make it work on your console.