this unit may well have an external resistor coil, and the resistor has gone bad.(sometimes these burn up when the key is left in the on position and the unit not running) the resistor is bypassed in the start position of the key switch.
I have also seen this with the ground (negative side of circuit) of the coil case not supplying the negative side of the circuit. this can be if the coil is not tight in it's clamp, and/or the clamp itself not getting good ground, including from the strap from the frame to the motor. (too much paint of the frame can cause this) I would check and clean all the ground side leads, if for no other reason than all the other electrical related failures you have had.
there are also some of these units that over the years have had a break internal to the 12volt battery positive from key to coil wire (right near the coil) where the insulation is still OK but the metal of the wire has broken due to "sympathetic vibrations" because the motor and frame do not shake at the same speed, and the wire is what runs from the engine to the frame, try pulling the wire to the coil, if it stretches, then you have a break internal to the wire where it stretches.