If you hot wire the ignition by connecting a jumper wire from the battery positive directly to the coil positive you will eliminate the primary wiring, solenoid, ballast resistor, and ignition switch. If you haven't already disconnect the tach. That leaves the Accel 2020 and coil which is enough to run the ignition. if it starts you will have to remove hot wire from the battery to shut engine off'. Make sure it's out of gear hand brake on and wheels chocked.

We both touched on this topic to sniff around the solenoid. The OP jumped the car with a known good battery and did all the tests per factory service manual for the primary ignition which were completed . Just thought it would be simple to try the points ignition and see the results.

Voltages north of 13 are typical for a battery holding a "surface charge", after recent use in a running engine. Standard lead-acid, the old fashioned kind with filler caps for watering, have an open circuit voltage of 12.65 volts at 77° F.

"Maintenance Free" 12.80 volts. If his battery is the latter type it is getting low.

Still requires 12 Volts from Solenoid Terminal to initiate starting the Motor. 6-7 Volts won't do it. Solenoid no good I think.

  

Every one that's trouble shot automotive electrical problems develop their own methods that work for them but are confusing to others. To level the playing field and start from ground zero I have to agree with Gene's previous suggestion that the points ignition should be re-installed to see if the car is capable to run.

The "I" Terminal should have very close to 12 Volts. This Wire goes to the + Side of Coil for starting. You only have 6-7 Volts apparently. Not enough to start the Motor. Are You using a good load tested Battery? Under cranking load it should still provide 12 Volts to "I" Terminal. If the Battery is good then maybe the Solenoid is the Problem.. Battery Voltage of 12.4 is very low for a fully charged Battery. 13-13.5 is more realistic.

  

I'm a little unclear on your cranking voltage, though measured directly across the battery posts 9.6 volts is considered a defective battery. There is a temperature correction factor to be applied, but a strong healthy battery should maintain well north of 11 volts while cranking.

On the start side of the solenoid I have around 9v while cranking. On the "run" side of the solenoid I have 6-7 volts. Battery positive to negative is 12.4. Did not improve while jumping the battery from my daily. I have a good ground at the breaker plate to engine block, good from engine block to battery. I don't have continuity from the negative on the coil to the distributor housing, not sure if that's right or not. Otherwise the wiring passes all of the tests in my 64 shop manual for proving out the primary ignition circuit. 

I had a similar problem on a 55 Thunderbird.  It left me stranded a couple of times and was hard to impossible to start.  After some testing, I found that the hot ignition wire was loose on the back of the switch.  A simple tightening of the nut and problem solved.  

Mac,
The factory wiring by-passes the ballast resistor when cranking the engine to give the ignition a boost to help compensate for the voltage drop from the starter motor. It does this with a terminal on the starter solenoid that connects battery voltage directly to the pos. on the coil. Did you alter this wiring ?