Error 1437, drive is good but not accessible to PC

I have a model A1136 iPod (5.0g 30GB). My trouble started when I got home from a trip and plugged the iPod into an iPad wall charger. After several hours on the charger, I disconnected the iPod and saw that it was alternating between a "sad iPod" screen and "battery low" screen.

Sometimes I could get it into diagnostic mode, and disk mode, and even "connect to a computer / use iTunes to restore". In the latter case, iTunes would spend a long time before it found the device, then would give an error 1437 "could not restore" message.

I replaced the battery with a new one and got the same behavior.

I replaced the logic board with a used 2006 logic board and got the same behavior.

I replaced the drive cable and got rid of the sad iPod / low battery cycle. However, I still can't get iTunes to restore the iPod. It shows up in My Computer (Windows 7) as a removable disk, without a disk inserted. When I try to format, it fails (no capacity is reported in the format screen). When I try to open it, Windows reports that the drive is inaccessible, "Incorrect function".

When I boot the iPod to diagnostic mode, it does report the hard drive model and capacity correctly, and displays the SMART statistics which seem quite normal.

When I connect the iPod to a Linux machine, the USB device is found and recognized as a drive by the SCSI subsystem, but then after several timeouts the system determines that it is inaccessible.

The drive works fine when I connect it to the PC using a USB drive enclosure for iPod drives.