Generator craped out?

Paul, I always just figured you'd have an ONAN BFA or something in that series, but still ONAN. I say that because nearly all of us have ONAN followed by Generac then a few others including Kohler. We had a BFA, and it had a series of what I'll call Safeties or Health Checks that were disabled in Start Mode of the Control Switch. For example, our BFA, in Start Mode, didn't care if it had Oil Pressure and if it was producing its own 12VDC and 120VAC outputs. When the switch was released into Run Mode (the center position that the switch would spring back to from Start or Off), then the genny was on its own. Had to have oil pressure, Had to be putting out 12VDC, and I believe it Had to be putting out 120VAC. It accomplished all this with a network of switches, relays and fuses.
Nowadays, probably since the 1990's, the ONAN gensets use a Logic Board that's pricey, but likely causes less false shutoffs than all the relays and stuff the older ones had.
I can only imagine your older Kohler uses Switches, Relays and Fuses like BFA did. Suggest you:
1. Disconnect your interior control panel and see if it acts differently when you start it from the genny's on panel.
2. Check any Fuses and replace with known good ones for troubleshooting.
3. Do a thorough visual inspection. You may find a wire loose or off. I just saw where some of the panel screws in older ONAN came out and fell into the contacts. You may spot a problem.
4. Join smokstak.com. They have a Kohler section and people who work on generators.
Wish I could help more.

I don't know about Kohler, but some of the Onans have a wire wound resister in the circuit that can blow and cause the problem you are having. What happens is when you hit the starter it bypasses the resister to give full voltage when starting. Then when the starter circuit opens it gives a reduced voltage through the resister. If the resister opens so does the run circuit.

Don't know if this might help, but I had a similar problem. Changed everything then noticed the switch on the generator was vibrating and sliding to the middle not completely off.Iput a piece of tape over it and it has worked ever since.

Had this generator and it's tank sat in storage for an appreciable time like most used RV's that are sold? If so, you may have bit of residue in the fuel system.