I'd have to trace out the wiring diagram, but I believe that circuit works by having a hold current sent to the solenoid when the switch is on, and in the start position it sends voltage to the pull side of the solenoid as well as engaging the starter.
If it doesn't pull and hold when you turn the switch to start, I'd test the solenoid with a jumper wire from the battery to the pull pin on the solenoid. If it pulled in, check the voltage coming from the ignition switch. Chances are there is just a bad connection and cleaning the contact surfaces often fixes the issue.
