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The plugs should be gapped to .025", not .035". This is probably a lot of the problem.
The points should be gapped to .016"
This will not be a voltage regulator, it only controls output voltage.
There is no circuit board on this engine.
It should get voltage from the start switch ignition circuit.
Voltage should be around 12V, but 9 or 10 is usually all that is needed.
The auto idle will have nothing to do with the quality of the spark.
Likewise, the starter solenoid will have nothing to do with the spark, either.
If it has been awhile, it sure would not hurt to change the points.
But at a minimum, give them a good cleaning and a thorough examination.
If you are sure the coil is good, the 3 most likely suspects here are:
Bad plug wires.
Low voltage going to the coil.
Try jumping 12V directly from the battery to the positive terminal of the coil and see if this makes a difference.