I will say, while I agree with the broad strokes of what you're saying, in my experience when you provide an LLM with data to summarize, instead of it needing to go find it, the hallucination rate goes down staggeringly. I've never had a huge hallucination when the material I'm asking to be summarized is provided to it at the time of the request.

Any further distance than that though, even summarizing the conversation it's aware of up to that point, is dicier.