Handlebar switch runs the starter relay. Starter relay runs the starter solenoid. Starter solenoid runs the motor. Old wires is BS. Bad/corroded connections on the end of the wires is a maybe.

Where's the "click" coming from? Big click/clunk from the solenoid? Little click from under the seat area? I recentely had to rebuild my solenoid. It would just click sometimes. The contacts were worn. The Accel high-dollar chrome replacement was crap and made it worse.