They "recognize" mistakes just fine because you explicitly tell them. They recognize them well enough to correct (...sometimes). The way in which mistakes don't register is "Oh shit, that bad result was a result of my inappropriate actions. I must pay attention to not doing that again or the user will think I'm an idiot. I should even think about it some more to avoid the whole class of mistakes". Think of emotions as an attention mechanism that LLMs lack.

I personally think that the "inner voice" is a non-falsifiable claim, and therefore more of a religious belief than something which can be part of any materialist theory. In this regard, I'm a strict empiricist and wouldn't be able to claim that I have one myself. In fact, I find that thinking "out loud" or "on paper" produces much better results in most instances, probably because I'm grounding my thinking in natural language, which is a fantastic medium for thought. If my "inner voice" were comparable in efficacy to actually speaking or writing, we wouldn't notice this effect, but I'm definitely not alone in this regard.

Your point about writing and social intelligence is, to me, more evidence for the "it's language that's smart, not us" hypothesis. We start off in small bands of hunter-gatherers that store their intelligence in an oral culture. Language then jumps to clay tablets, papyrus, codex books, etc. The printing press allows it to escape containment to a wider public than just a caste of priests and bureaucrats. As soon as we invent automatic calculators, we start networking them and using those to process language, albeit in a primitive way (email, the web, etc.). Recently we discovered some abstruse math that, with the assistance of a bunch of beefy video cards, can crunch centuries of human writing into a mathematical object that encodes at least some of the meaning of that writing into an even more "advanced" symbolic processing machine. There's a clear trajectory of language itself getting more and more free of the specific wetware it grew up on.

It's a falsifiable claim, in that if there is a way to train a useful LLM from scratch without any human authored input language to bootstrap it (something I've been on the lookout for but haven't seen, though admittedly I'm not an AI researcher, just some Linux nerd with a day job as an SRE), then we can disprove it.

For the religious angle, look no further than John 1:

"In the beginning was the Word, and the Word was with God, and the Word was God."

(This is admittedly less falsifiable!)

Yes, this to me is also a good sign for the "it's language that's smart, not us" argument. It's an emergent trait that has evolved several times, like flight or carcinization. There's something about language that attracts evolution toward it. One would expect such a trait to have a big survival value (disclaimer: IANA biologist, philosopher, theologian, mathematician, or linguist).