Not that I have any special clues here, but I sure liked Hillbilly's answer as the best fit to the randomness and exclusion of any preconditions for the no start. A short brush or bad spot on the fuel pump's commutator would 'splain it all, except for your very thorough report of its humming away nicely while not starting.

Failing my test for elegant solutions, I agree with your next course of action with the ignition module. The problem with all of that great spark you see is it may missing when you need it. Kinda hard to test the spark and start the motor at the same time, though not impossible.

If you only could depend on it not starting, you could use a timing light or the radio set on AM at the low end of the dial, or a neon lamp with some leads wrapped around a plug wire, to assure yourself there is spark when it is not starting. Then you could replace the coil Patrick suggests, although intermittent coils are usually influenced by temperature and moisture in their fickleness.

I have a 79 that uses a ballast resistor in series with the coil (near the right hood hinge) that is taken out of the circuit (bypassed) by a contact on the starter solenoid. I've never had a problem with it, but I wonder how much harder to start could it be if the solenoid contact was burned, or the wire got loose or fell off the starter, preventing the ballast resistor from being bypassed. The symptom might be an intermittently weak spark while cranking, but running would be fine. I wish you were pouring concrete in my back yard.

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Art Benstein near Baltimore