That seems like an extreme interpretation if I'm honest. As ever, its more likely somewhere in between your hammer analogy and the handyman. In any case, there is almost zero downside to listing your sources when answering a question as it provides the person asking the question with more context as to the origin of the answer
If you provide sources, provide the sources the LLM found for you after checking them yourself. Otherwise the recipient will have to vet them.
Slight rant: I’ve started to get a bit fed up with people antrhopomorphizing these things, it’s unhealthy. It’s definitely not a handyman and it really is just a tool. I know they’re designed to seem conscious and that just makes them dangerous to use. If my colleague starts treating his hammer as a person, it’s definitely not normal.