I suppose the LLM doesn't know it's limited in its knowledge, maybe? That others know more.
If the training data doesn't include lots of text of people being asked questions and saying "I don't know", then it's unlikely to respond "I don't know" when prompted, regardless of whether anything in its training data that might actually answer your question.
There's a problem with your reasoning. The above applies to a foundational model (that is to an autoregressively pretrained model). If the training data doesn't contain "I don't know" in a dialogue context, then, indeed, the model is extremely unlikely to output "I don't know" when asked. That's the nature of autoregressive training.
But we are dealing with fine-tuned, RLHF-, intruction-, RL-trained models. If the model has "grasped" a concept of knowledge, this concept can be elicited during the mentioned training.
Humans run on training data too in the same broad sense.
It'd be interesting to see if distillation increases hallucinations for specific topics the larger LLM is confident in