Starter
I have a C27 and the starter circuit is rigged as follows. When the ignition switch is on, power is on to the starter button. You apparently use the start position in the switch - works the same way. When the start circuit is energized by pushing the button or turning the switch to start, current flows from the button (or switch) through the wiring harness to the S terminal on the starter solenoid. On the C27 this is a 14 gage white wire. Your problem occurs because there is an intermittent open circuit somewhere along this path from start switch to solenoid. The fact that it will start if you charge the solenoid start terminal directly tells you the same thing. Since you have a new starter switch, the open circuit is somewhere in the wiring. The simplest solution is to run a new wire from the start terminal on the switch to the start terminal on the solenoid. Better would be to track down the problem (probably a bad connector) in the wiring harness and fix it.There is another, less likely possibility. That is that the current to the switch flows through a loose connector somewhere which will conduct with the normal loads but not when the start circuit is energized. Since the current to the solenoid isn't very large, this is probably not the answer. I just cured a very similar starting problem last weekend caused by a loose ground connection. The blower would run but everything shut down when I tried to start the motor. After a while the blower would mysteriously come back on. After much cussing I tracked it down to the starting battery-to-engine block ground connection at the battery terminal. Tightened the wing nut and everything was fine but two hours lost.Good luck