power to the neutral safety switch is great, but coming out of the switch is the important thing. If you have 12V + coming out of the neutral safety switch, then check for it at the round connector at the engine bulkhead. The yellow/red 16 ga. wire supplies power to the "assist solenoid" aka start relay for cranking. From the high current terminal, another red/yellow 12 ga. wire powers the starter solenoid.
If you have power to the assist solenoid but no click, then solenoid could be faulty, or it could have a bad ground (I think it grounds via the body of the solenoid to the bracket)