There are two coils behind the flywheel. One supplies the timing\voltage for the spark, and the other supplies the voltage to charge the battery.
You say it runs until the voltage gets too low -
It should run all day, even with a dead battery. the starter would quit, but the battery is not used for "running" the engine. The battery is only used for the starter.
The above statements apply to a stock configuration.
There is an aftermarket flywheel system known as a "total loss" system.
It doesn't charge the battery at all. The advantage of this is a much lighter flywheel, which gives increased performance.
So if you are sure you have the stock system, then it is probably the coil under the flywheel.
It could also be the voltage regulator. That is a small metal box inside the electrical box. It is in charge of regulating the voltage to the battery.
One thing I recommend to everyone who has these old JS skis, is to run nice thick ground straps between the engine and the electrical box, and from the electrical box to the neg terminal on the battery.
These skis were built using a tiny wire to provide ground for the system. I had a lot of electrical issues until I did this. I haven't lost a component since!
OL JS Rider is correct about a coil sometimes passing a resistance test but failing under operation.
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Everything can use some modification.
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