I’ve never dealt with one of them, but I’ve had that issue. I’m a Mopar guy and up until the late 70s, old mopars had a ballast resistor. It had two circuits,, one for engine crank and one for it to stay running. When they go bad they will have the same problem you have, only have spark while cranking. Those style systems are so the full 12 volts goes to the coil for better spark, resulting in easier starting. One you let go of the key, voltage drops down. I’m assuming your situation is something similar.
Currently : trx400fw, ct70, 250es,200x, xl80s, 350x, 350x, trx300, trx300fw