Actually, on the 2nd Gen Nano, none of the buttons are on the logic board (see step 20 of the teardown). The clickwheel is an entirely separate part, and just a cable runs to the logic board for all the click/scroll input.
The clickwheel is your starting point to replace.
Unfortunately, this iPod was created when Apple started getting a bit adhesive-happy. The click wheel is held in place by a metal bracket stuck to the casing with strong adhesive. If you endeavor to replace just the click wheel, make sure you have a STRONG adhesive handy, or else the click wheel will just collapse into the iPod when you push it. If you replace the case assembly, make SURE the place you get it from includes the click wheel attached to the case assembly.
If replacing the click wheel doesn't fix your problem, all signs point to logic board, unfortuantely.
I hope this helps!