If it works everytime with the bypass wire, I would say you have a switch or wiring issue.
GM's typically use a purple wire form the Ign. switch directly to the solenoid to energize it.
When the solenoid energizes it pulls a shorting contact forward that send battery power to the starter motor and on PRE HEI cars battery power to the coil bypassing the points resistance wire for starting. It also sends the Starter Bendix back to the flywheel engaging the teeth before the SM starts turning.

Energizing the solenoid is audibly noticeable. So while it is possible the solenoid is at fault, I would suspect the switch or wiring first.