I think this mostly hinges on whether you interpret Jaynes' argument of the origin of consciousness as being a cultural artifact or an evolutionary trait. We usually think of it as the latter and tying it to a certain recent point in history in the West suggests a perspective of people outside that sphere as "less human" based on how we usually think about consciousness.

I think Jaynes' actually seems to argue for consciousness as a cultural construct. While this makes sense in a way, it largely conflicts with how consciousness is commonly understood, especially in the context of moral, ethical or religious arguments.

I'm also not sure to what degree the "bicameral mind" would have been a universal experience. While self-talk is extremely understudied we now at least know that some people have inner dialogues (which we usually consider a form of hallucination or schizophrenia), most have inner monologues and some have no active inner voice whatsoever (if we ignore subvocalizations and only focus on internal "thought"). In religious contexts we still often treat ancient stories of people "hearing the gods" at face value even if it is no longer culturally acceptable to claim the same today (i.e. you'd rightfully be considered to be suffering from hallucinations). I'm unconvinced the examples Jaynes gives are more than just artifacts of cultures that treated hallucinations as a meaningful part of the human experience the same way some cultures (including people as recent as Freud and Jung) treat dreams as meaningful parts of the human experience or religious practice.

Also externalization of intrusive thoughts or emotions is actually part of therapeutical approaches to disorders like depression. Even meditation and mindfulness often involves viewing the inner mind as a detached observer. I'm not convinced attributing these uncontrolled experiences to an internal "id" (with full acknowledgement of the caveats that come with referring to Freud in the context of modern psychology) is much more sophisticated than attributing them to external "gods" or demons. So in a way the "bicameral mind" still seems to exist, except that we have changed our cultural narrative around it to one that favors the fiction of individual responsibility over the fiction of supernatural forces outside our control.

I think the only reason the book has been so influential is that people misread the argument as an evolutionary one rather than a cultural one and conflate Jaynes' meaning of the word "consciousness" with the popular and religious concepts of it. I also think this mistake is particularly appealing to those already looking for reasons to justify treating "Europeans" (i.e. white people) as biologically superior to others as it neatly fits into the colonial era racist idea of the "industrious Asians" only being intellectually superior by having a "hive mind" mentality and being individually empty and lacking original thought (and likewise Black people being too primitive for complex thought and thus stronger and more violent). I'm not saying Jaynes would agree with any of these ideas, I'm just saying people looking to justify these ideas (whether or not they're willing to openly state them in public) will happily misinterpret Jaynes' work to do so.