Check your game patchs, All fails fresh install of FSD if that fails you may have a HDD problem Had over 300 games on external hdd last year and lost the lot. It started off the same (games not showing) in the end it was the hdd it had gone. May also wanna try xexmenu see if there showing and boot fine in there.
Try the external hard drive in a PC, if it finds it the drive is fine if not you need a new drive. If the PC finds the drive do the following: Try resetting the FSD databases. Boot into XeXMenu 1.2 Locate your FSD folder > Databases > delete those two files. Re-boot and reset your content folder(s) with your arcade, 360 games and make sure for 360 games the 360 Games option is checked for both check boxes and Arcade is set XBLA in the check boxes on both sides.