From emails to the ipodrepairs.co.uk, it's possible the battery drain issue is
probably a problem with the LTC4066 power management IC. Upon full disassembly of the iPod, there is visible corrosion around it, as well as around some areas connected directly to the 30pin port. By corrosion, I mean green/blue copper sulphate looking stuff or white crystalline. It is minimal, but the components are tiny. Maybe if I'm lucky, cleaning that off with a bit of isopropyl alcohol will make it work properly.
From my reading, it is likely a complicated fix otherwise. Other people cut traces, soldered diodes, or even suggested buying
a new LTC4066 component, but replacing that is likely impossible due to it being surface mounted, with a blob of heatsink solder behind the middle of it and a bunch of small caps/resistors nearby. And I don't have a hot air station.