What model of Onan 9k?
Many have a 30 second delay on the shutdown circuit. So, it could be, for example, the low oil pressure switch/circuit, or the 120v output check circuit, or, as you are guessing, the high temp switch/circuit.
I'll feel better commenting once I can look up the schematic, but normally the safety switches alarm by shorting to ground. So, if you very briefly disconnect the wire for each safety switch and keep it insulated from the block and ground, you can see if it runs longer. Of course if it is overheating or low-pressure, etc, there is an obvious danger.
Personally, when I do this type of thing, I like to put a mechanical gauge on oil pressure and measure temp with an IR temp gun. They are cheap at Harbor Freight, Amazon, etc, and it let's me verify that it is a false alarm vs real problem.