Many of the new replacement switches have all three positions, i.e. ACC, RUN, and START. On my mechanical foot start, the original switch was only 2 position. The new switch looks identical but, it has the START spring loaded position. It is not used.

One thing I would ask, did this vehicle get a 12 volt conversion by any chance. Does the motor have the original points ignition? "RUN ON" can be the motor running off of its charging source. This is a common thing with ignition swaps and alternator swaps. It is fixed with a diode in the "field" wire. The next suspect would be the ignition switch itself. To find the culprit, disconnect the "Field" wire with the engine running and the key off. If it stalls, it is feedback from the regulator. If it doesn't there is 12 volts to the positive side of the coil from somewhere, most probably a bad start switch.