Sounds like [http://discussions.apple.com/thread.jspa?messageID=7071017|this guy] had the same problems that we are having. Like you, he concluded that the issues are NOT related to the hard drive. The [[iPod Mini 2nd Generation Troubleshooting|iFixit Troubleshooting page]] for the 2nd Gen iPod Mini suggests replacing the hard drive CABLE. Not sure if you have tried that or not, but it may be worth a shot since it's much cheaper than a logic board. Perhaps you could take the hard drive cable from the working iPod and put it in the bad iPod to see what happens to avoid spending money on a new hard drive cable.
If the hard drive cable replacement doesn't work, it's certainly a logic board issue, as you originally said.
It seems to be impossible to make the iPod do anything besides a reset. I too have been unsuccessful in restoring, disk mode, or having the iPod recognized by a Mac or PC.
I'll probably purchase a hard drive cable in the next few weeks and I'll let you know what happens.
Keep me updated, please. Best of luck!