By the way, "not turning over" means the motor does not crank. If it runs at all, even for a few seconds, it definitely is turning over.
It is worth your time and money to troubleshoot the generator. A replacement Onan will be a four-figure expense unless you get a used one. Besides the oil problems alluded to already, there are a couple other things that will make the controller shut down the generator quickly, such as not seeing proper electrical output.
It may also be that you're just not priming sufficiently. It can take a fair bit if the gas line between the tank and the generator is long, maybe 20 or 30 seconds or so (which feels longer to me when holding a switch down).