Your 4KY spec K does not have an oil pressure switch nor an oil level switch. The engine is splash lubricated so no oil pump and the oil level switch was done away with in a previous spec.
With the engine starting and not staying running, I would suspect the voltage regulator circuit or the brushes and slip rings/rotor assemblies. I would start with the regulator which is built into the control board. If you can, find the two wires that go to the brushes and slip rings. Apply 12 volt to these wires and if the unit stays running, you have a control board/regulator problem. If the unit still stops, place a multi-meter on the same wires and check the service manual for the correct ohmage reading to determine if the rotor or the brushes are bad.
Do not attermpt to remove the flywheel and repair unless you have the service manual. Lots of small parts and sequential disassembly procedures.
JimL