In the spiritual teachings & traditions the wording would be different. I think they would refer to was Jaynes call "consciousness" as "self awareness"; and then assert that "consciousness" is prior to any of that. For example in the dream state we are typically not in control, nor are we aware it's a dream.. yet when we "wake up" we remember having dreamed and having "been there".
In fact I'd argue that we DO many times of the day every day... speak and act without much self awareness. And yet, we remember afterwards and then claim those moments as something we did.
> In short, Jaynes claims that men in the age of the Iliad learned to speak, read, and write, as well as conduct their daily lives, yet remained nonconscious throughout their lives.
Maybe they were indeed much less "self-aware" as we are today, which again is not same as saying not conscious.
edit: it seems to me this is the main contetion in science often times people just conflate mind activity and consciousness , as an (possibly not deliberate) attempt to chuck away the "hard problem of consciousness".
pps: funny enough it is actually an hypothesis, that for human beings to recongize consciousness is also something made possible through self awareness. For example, what is called "awakening" in spiritual teachings is supposed to be the natural state, hence, whatever "spiritual" state it is thought to be, is a self-refetential acknowledhgement of a change of one's perception of self and world. That is, "self awareness" allows for human beings to acknowledge consciousness (since we have to conceive of it).. whereas a dog or cat can be consciouss, and relatively free of worry, yet will never know themselves as "free of worry" (I mean in a rational / self aware way).