It would be cultural and taught, not genetic.
There are non-isolated populations: I come from a religious background where the idea that God "talks" to you is quite prevalent. If you want an analysis of how could possibly be the case in our modern society, I recommend Tanya Luhrmann's "When God Talks Back".
I say Jaynes puts the transition as far too sudden compared to what it actually could have been. Individuals could have had developed his version of "consciousness" often and repeatedly, but failed to pass it on into greater culture. I think of literacy as an analogous transition; until it's actually of benefit for the average person to read+write (ie society has moved beyond majority subsistence), it's not going to happen.
[0] https://www.goodreads.com/book/show/12958803-when-god-talks-...