There is no possible way this has anything to do with the ignition coil, ignition module, or anything else in the ignition system proper.
We're told that the solenoid clicks. Therefore the ignition switch, neutral safety switch, and starter relay are working. The purple wire between the relay and the S terminal of the starter solenoid is carrying at least some current.
The 95% likely fix for this is to replace the starter and solenoid. There was a time when replacing only the solenoid might take care of the problem--but nobody replaces just the solenoid any more; and at a hundred-forty thousand miles, I wouldn't keep the starter after going to the trouble of removing it to scrap just the solenoid.
I'm curious about what type of failure a "dead spot" is, mechanically speaking - does that mean there's a broken/worn tooth that is preventing the engine from cranking on when it lands on that spot, or is it something else?
There are two possibilities for a "dead spot". One in the starter solenoid--there's a relatively large copper disc that gets pushed into stationary copper contacts. The disc is free to rotate; but there's no mechanism that forces rotation. If the copper disc or the contacts it's pressed against develop a burnt spot, the starter drive gets shoved into engagement--but the motor gets no power so the starter motor doesn't turn. Click!---then nothing happens. Just exactly like what you have now. Maybe there's enough vibration to turn the copper disc a little--now the burnt spot isn't aligned with the other contacts, and the starter motor works normally.
Photos of ancient, easily-disassembled old-style starter solenoid:
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The starter motor has a certain number of armature bars. Each bar is connected to armature windings. Electric power flows through each pair of bars in turn as the motor spins. If there's an open in the wiring between a pair of armature bars, those two bars are "dead". The starter motor may spin fast enough that those dead armature bars don't contribute to starter motor power; but the motor still spins with reduced power because all the other bars are working. However, if the motor happened to stop spinning with the dead armature bars aligned with the brushes; the starter would never begin spinning.
In one situation, you'd need a new solenoid; or at least polish the copper disc and it's stationary contacts to make it conductive again. Nobody does that--it's not economically viable if your labor is worth anything.
In the other situation, you'd need a new starter armature. Almost nobody fixes starters piece-by-piece any more. Again, it's not economically viable if your labor is worth anything; and it's hard to buy just a replacement armature.
So either way, you're trading off the starter/solenoid for a new or rebuilt starter and solenoid.