For what it's worth, I just replaced the starter button on the engine control panel today. Over the past month you really had to mash the button down to make it work, then it finally failed this weekend. I actually started the engine using the hand crank with the compression reliefs open. The new switch cost $30 with tax, and works like a charm. It gave me another idea for troubleshooting your problem.

There is a connector at the aft end port side of the 2GM that joins the harness to the engine control panel. If you unplug this, the two fat lugs are for the starter switch, and all the others are for the sensors & idiot lights. Try unplugging the connector, and on the engine side take a pair of needle nose pliers to jump the two fat lugs. This should engage the starter, through the starting solenoid. If it works, then maybe you have a bad contact somewhere upstream to the push button on the engine control panel.