Both versions, actually. The Cerebral cortex does have a lot of memories, etc, but actual consciousness is an emergent phenomenon that is more like a subroutine skimming functionality from the overall neurosystem flows, and motivated executive function rides on that,
https://pubmed.ncbi.nlm.nih.gov/33997292/
That's why anesthesia works to shut off consciousness without destroying memory (usually) or killing us.
We also should keep in mind that much of the functionality is tied in with the overall systems and movement of the whole body. Recent information suggests that our ability to compose language evolved from our toolmaking, rather than social development (cart or horse, but toolmaking came before sapiens).
It's important to wonder how much of society is really intentional, and changing may require embracing our 'lizard' for good, rather than believing education and legislation can beat desire-based traditional exploitation.