When it won't restart, I assume that it cranks. If it cranks and will not start, that would have me back thinking maybe a fuel system problem. I have had a situation in a boat where a fuel filter would load up with contaminants on the filter face essentually plugging it and killing the engine. With time, the fuel pressure would bleed off that held the contaminants on the filter face, unplugging the filter face and the engine would start again and run for some time only to repeat the process again. I have no way of knowing if that is happening here, but it may be possible. Seems like you could crack the filter output side fuel line when it shuts down and see if fuel is present while cranking, given it as a relativley simple check.
The fuel cut on/cut off is probably a 12v feed to a solenoid valve, or an electric fuel pump. If that 12v feed is accessable to get a measurement, you could check the voltage while first cranking and running normally. Then see what it is when it quits. Then see what it is when it cranks and will not start (assuming it cranks). If while cranking and not starting, the voltage is more like it was when it was cranking and running, then I would think the fuel system is at fault and must not be delivering fuel. (air in line, plugged filter, etc) If the voltage was more like it was when shut down, then I would think that the control board is not enabling the fuel pump or solenoid valve electronically, due to a sensor or control board issue.
Later Ed