I don't think it needs to be external to the natural world at all, nor is Jaynes trying to "solve" consciousness in the typical philosophy of mind way, he's simply reporting a theory about the phenomenology of thinking in early human evolution, based on historical evidence. I'd agree that it isn't very strong evidence, but worth considering nevertheless. Regarding the connection with evolution of culture, because it doesn't line up with the evidence that Jaynes describes, the "switch" seem to occur much later. And it's quite possible that the cultural explosion allowed many aspects to emerge such as complex language, self-awareness, and abstract thought, without having anything to do about how "Gods" are experienced in the way that he's arguing.