Jeremiah,
The schematic is correct for the most part, just does not show the exact/detailed contact setup of the switch. When the key is in the start position there is still a closed circuit for the ignition.
Clark,
The spark occurring when the key is turned off tells us that the ignition circuit is somewhat operational, i.e. good coil and condenser. Turning off the key (opening contacts in the switch) is the same as opening the contacts on the points. The points are not performing as they should, why?
If the points are used/old clean contacts with a small CLEAN file. Slip the file between the contacts and move back and forth several times. Would not hurt to do on new points, just a few strokes.
With the key in the run position and the points in the closed position open the points with a non-metallic, non-conductive object, like a dry popsicle stick or such. You could tape up a screw driver with electrical tape. Gently pry on the plastic arm of the breaker point assembly. There should be a faint spark created when the gap gets large enough and then at spark at the plug, it will appear simultaneously. This will be hard to see in a well light garage or outdoors in the daytime.
If there is spark at the plug, then gap adjustment is likely the problem.
If there is NO spark, breaker point assembly is defective or grounding elsewhere.
Or the switch like Matt stated.
Jim