'84 CX650E that is evolving into a GL500
Dogmuck and Jpchi: He has it starting now but noot running well so the problem is not likely to be electrical.
Mine was the solenoid. I tried 2 cheapies from EBay, both defective. I finally resorted to adapting and making room for one of the old Ford style ones, much more robust. It was a real challenge to get it to fit under the side cover, had to jostle a couple of things around and grind off most of one terminal but in the end I feel much more secure in my old ride (81 CX500)I'd start with the button. Test for 12vDC at the input side of that switch, then test for 12vDC on the output side as the switch is cycled. A likely culprit is the starter solenoid, but it's best to start upstream and work your way down.