Aside from basic problems, most sets that vintage and caliber have a safety loop... a circuit intended specifically to shut the machine down in the event of some basic failure, which usually includes:

1) Engine coolant Overtemp

2) Engine oil pressure loss

3) Engine overspeed (sometimes mechanical governor switch, a centrifugal switch on the generator end, a frequency sensor on the generator, a flywheel proximity sensor, an output on an electronic governor, etc)

4) Low oil in oilpan

5) Low coolant in reservoir

6) generator overtemp switch

There could be many others.

These switching devices are always 'fail safe'...meaning,they have be 'maintained closed' by proper operating circumstance, and return to a 'normally open' circuit condition when the failure occurs.  In some cases, this is done using contacts of a protection relay.

In order for a generator to start,however, the safety loop must be bypassed for long enough to get the machine running and into a 'normal' state.  This is usually accomplished by a simple timer relay that activates when the machine is started up... And once it's running, normal conditions result in the safety loop being closed, so it can then take over after the timer drops out.

I would study the machine carefully, and find out what the control system has in it, specifically, what device controls the fuel cutoff relay, and also check all the fluid levels, gauges, etc., for hints. IF you hear a relay CLICK just before it shuts down (and I realize that when the 3500 is running, that's like listening to a mouse whisper at a rock concert...) then it's probably either the timer, or a control relay SHUTTING it down.

BTW... is your 3500 still diesel, or was it set up for spark-ignition NG/Propane?