If the downhill output side of the resistor has voltage on its input, but none ever on its output, there must be a short from ballast output to ground causing it to draw excess current constantly and yes overheat OR THE BALLAST IS AN OPEN CIRCUIT. On the ballast output, which is the coils input (NOT to distributor side), with ignition on and points closed Id expect around 6 volts there UNLESS there's a short to ground. Then over on the coils other output to distributor Id expect 12 volts when points are open, but zero ONLY when they are closed.
Look for a short in the circuit from ballast output which should normally be to and through the coil to the opening and closing points (in which condenser is a parallel path) then to ground ONLY if points are closed. If the coils LV primary is only shorting once warmed up that could allow it to run a while then die.
If the condenser is shorting out (could happen only after its hot ???) that would cause constant coil and ballast current (no opening of points to break the circuit) which will overheat and there wouldn't be any spark.
I would check ballast resistance when cold and warm,,,,,,,,,,insure points are opening and closing,,,,,,,,,,,check coils LV primary (+ to - terminals) it should be around 1.2 to 2+ ohms,,,,,,,,,,,,,,,,check condenser it may be shorted or shorting once hot,,,,,,,,,,insure the distributors pass thru stud insulator (where wire from coil attaches) isn't shorted to case.
Since the condenser can act up when hot and its much cheaper then a coil, try it first. However the coil may short only after its hot also. Coil and condenser are a couple prime suspects and sure the ballast resistor could have a problem so check that also, but there can be other problems, post back if it isn't condenser or coil and we will take it from there.
It obviously cant fire if there no volts getting to the coils input and cant fire either if the coil or condenser is shorting when hot, neither can it fire if the ballast became an open circuit (high resistance) but once it opened fully it should stop drawing current and cool off.
The coil and condensor and ballast aren't hard to troubleshoot, if its not the points or a wire shorting one of those three may be the problem
John T