This iPod charges, and syncs to iTunes. But when it is turned on, it displays a sad iPod icon. If it is turned on after...

Sounds like 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 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!