It does know what's accurate. Not only if you look at the logits, but it's also reified somehow such that if you ask LLMs how certain they are about an answer then they can actually express that somewhat accurately. This is why asking an LLM to avoid making things up can work (likewise, why asking a coding LLM to write secure code can actually cause it to not write security holes).

One reason LLMs "hallucinate" (lie/BS) is that there are subtle biases introduced by the training process. In particular human raters don't know what the model knows, so if the model makes up an answer that happens to be correct then that behaviour is rewarded. Also the internet doesn't have many examples of people saying they don't know, because people who don't know just don't reply.