The thing with the oil leak, when the genset is running, cooling air is being drawn through & around the engine, so an oil leak could be in one place & showing up somewhere else from the forced air pulling it.
Places it could leak, crankshaft seals or where a sealing gasket is located, part bolted onto the engine.
After dealing with these things for a few years, I believe a lot of times it is the rubber o ring gasket on the oil filter not sealing like it should. There have been a few times after a bit of genset use, once cool I've been able to tighten the oil filter up just a bit. Now that could have been from vibs causing the filter to be loose or either that o ring gasket srinking up, which I dont know, but that is one thing I have found.
If the genset oil pressure sending unit does not sense oil pressure, the control board will shut the engine down.
Neil