I try to get you guys to think of "the system". "The (end to end) path" "How does it work"
So here is what you have, end to end, the path.
The ignition switch, when twisted, provides power to the "start" contact, out the SWITCH, through the SWITCH CONNECTOR, and through the BULKHEAD CONNECTOR into the engine bay and connects to one of the two flag/ PUSH ON connectors on the STARTER RELAY and to the starter relay COIL---through the COIL and out the second PUSH ON connector to the dark wire, which goes on down the firewall, over the (automatic) transmission and connects to the CENTER TERMINAL of the neutral/ reverse switch. This SWITCH CONTACT is grounded in park or neutral.
Every CAPITALIZED terminal point is a possible point of trouble, along with the oddball broken/ pinched/ grounded wire etc "generally"
SO WHAT DO YOU DO?
First thing is to prelim test for neutral switch mis-adjustment or intermittent
Twist the switch to start. Might be good to pull and ground the coil HT wire to prevent the engine firing. Wiggle the shifter from park to neutral and back, and look for intermittent clicking or lack of at the engine.
Raise hood, in a quiet area, and listen carefully while trying to crank. IF the engine won't crank, LISTEN for the sound of the "small" click of the starter relay, vs the "heavy" click/ thud of the starter solenoid.
Since it always seems to crank if you jumper the solenoid WHERE ARE you jumpering the starter? You jumpering it AT THE starter or at the relay? If jumpering at the starter, it still could be the big fat wire "broken" that leads from the relay to the solenoid. This is the one connected to the bare "square" terminal with a screw. Those CAN BREAK down at the starter from vibration INSIDE the connector/ insulation
Try to eliminate the car wiring and determine if it is the starter relay itself. You can eliminate the neutral switch and wire, by simply disconnecting it from the relay--the dark wire"push on" connector which leads down to the transmission. Take an alligator clip lead, and ground that starter relay terminal. CAUTION!! Engine will now start IN ANY GEAR so be careful. If this eliminates the problem, "then you know."
Do same with starter signal from key. You will have to disconnect the yellow start wire at the relay, and start the car by jumpering from the "big stud" on the relay to the bare "push on" terminal you have disconnected.
IF THE RELAY CLICKS but the STARTER DOES NOT, then it's either bad relay contacts or a bad wire to the starter solenoid.
