Existing LLMs are already trained to mimic humans - by imitating text, most of which is written by humans, or for humans, and occasionally both. The gains from other types of human-mimicry don't quite seem to land.
The closest we got to "breakthrough by mimicking what humans do" since pre-training on unlabeled text would probably be reasoning. And it's unclear how much of reasoning was "try to imitate what humans do on a high level", and how much was just trying to generalize the lessons from the early "let's think about it step by step" prompting techniques.
It's likely that we just don't know enough about the human mind to spot, extract and apply the features that would be worth copying. And even if we did... what are the chances that the features we would want to copy would turn out to be the ones vital for consciousness?