Let's think this through:
Ski will start and run with button pressed...means solenoid connection is made, current flows through solenoid to spin the starter, the starter spins the flywheel, the engine is pumping, creating vacuum to pull fuel into it through carbs. Spark is generated to keep the pistons pumping. The stator is spinning and should be creating voltage to the Rectifier which feeds DC volatge back to the battery (if all is working as it should). The ECM should be managing all of this, in a perfect world.
Let go of button, ski stops...huh? Letting go of the button should drop the starter out of the equation, the starter did its thing so we don't need it anymore (meaning the solenoid should drop out too as it did its thing to get the ski started.) The combustion process should be maintaining itself with fuel, air, spark and timing now.
OK...think F.A.S.T. (fuel, air spark, timing). The fact that she starts and runs sorta rules out F, A and T. I think you are losing SPARK somehow to keep the engine going.....could be the induction pickup coil...BUT the ski runs holding the button which I think is forcing the starter to keep the flywheel spinning (so stator is spinning which the pickup coil is looking for) to keep the combustion process going by force.....hmmmm.......could be the ignition coil itself.....is the coil not getting its signal from the ECM with the button released (but releasing the button kills the mechanical process of turning the engine over thus dropping out the signal (so I'd rule that out)...hmmmm....bad grounds, wiring, corrosion......start checking everything in the harnesses...this sounds ELECTRICAL in nature....IMHO. It's a '95...she's old.