Instead of starting an identical thread: I have a 64GB 3rd gen exhibiting the exact same symptoms with one exception: when I plug the iPod into my Sony clock radio's dock, it comes on (cold boot) and everything works. Unplug it and it totally powers off (even thought the battery reported fully-charged). Plug it into Apple power adapter and the same thing happens. It's fully functional (except no wi-fi at all; it's grayed-out in Settings). I can play music, videos, games, anything. Home button works, power button works. Everything seems just peachy. But plug it into computer and I get bupkus. Leave it unplugged and also nothing. Home button doesn't work, main power doesn't work.
Note: this iPod went thru a complete wash cycle 21 June last year. It made it out of my jeans pocket to the bottom of the washtub. After "soaking" in in rice for about a week, it still showed no signs of life, so I took it apart to see all the pretty guts. It has been sitting disassembled since then.
I reassembled it yesterday, 26 Feb, and got a lovely surprise. I'll try the suggestions re checking power, making sure everything is clean. Failing that, any other suggestions? or should I replace the logic board?