The part where we can "analyse what we just said (or were about to say) (or what we just typed and didn't press send yet!) and conceptualise it as a whole to evaluate its veracity" involves going back-and-forth in yourself. A loop. LLMs are the part inside the loop, and so if you want to get better results, you have to feed the output back to LLM.
This is, arguably, part of what the "conversation mode" does, by always including the growing message log in each request - which is why e.g. GPT-4 is good at correcting a mistake once you say, without pointing it out, that there exist a mistake.