Besides the scrolling feature built into the click-wheel, there is a microswitch button for each function. In other words, Menu, Forwards, Backwards, Play/Pause and Center Button each have their own switch that’s activated when you press that part of the wheel.
It’s possible that the switch for the Center button function has failed. Other than making you reload all of your music, restoring won't hurt and it COULD reload something that MIGHT have gotten corrupted on the iPod, but I wouldn't have much hope that restoring would work.
Parts, like click-wheels are available on eBay and other places and there are places that offer repair services for ipods.
This might be a good time to start looking at iPods on eBay.