The rust and trash in the fuel cutoff solenoid can physically jam the solenoid and prevent it from working properly.The seat switch is a common problem.Put a jumper across the wires of the switch and see if that solves the problem.If so,replace the switch for safety.
Kohler had lots of problems with the DESAI ignition,and finally abandoned it and replaced it with the old fashioned grandpa version ignition system..back to the old reliable.
All the DESAI system did was delay the firing of the plug.The initial timing was set high, and the delay was long,so timing was low at starting, to allow easy starts,and as the engine increased speed,the system decreased the time delay on the timing,thus advancing the timing.If the battery is border line,you may have trouble starting due to the high timing with the old fashioned coil.AIso, advance the coil a bit by loosening the screws,and moving it in the direction of rotation as far as possible,thus reducing timing, then resetting the gap.There is not much clearance in the coil screws,but a lttle goes a long ways on timing.A vaccum advance would have ebeen much better,IMHO,like in older cars.
There will not be a noticeable difference in performance from my experience.