I'd argue they don't synthesize any ideas, even old ones. They skip that classic step to emit text, and the human reading that text generates their own idea and (unconsciously, incorrectly) assumes there must've an original idea that caused the text on the other side.
So perhaps it's more like: "LLMs aren't great at indirectly triggering humans into imagining useful novel ideas." (Especially when the user is trying to avoid having to think.)
Yeah, I know, it sounds like quibbling, but I believe it's necessary. This whole subject is an epistemic and anthropomorphic minefield. A lot of our habitual language connotations and metaphors can mislead.