How would you normally check a disk volume for errors on the Mac? However you do it, do the same with the disk that mounts from the Gen3. Eg, on Windows, you right click on the disk icon, select "Properties", then "Tools", then "Error checking". Presumably the Mac has something similar?
From your description, however, it seems clear that you do have a corrupt internal file system. What I'd do is make a backup of the "system" folder, then reformat the internal disk (to FAT32) using your Mac, and put the system folder back. That should fix the problem.