Am I just being boringly cynical when my first reaction is "Uh, what else did you expect"?

ChatGPT is not... a computer. It's not a human. It's not an intelligence. It's a way of turning small amounts of words into large amounts of words in ways that is sometimes extremely useful.

It's not a logic engine. It doesn't draw inferences or prove points, except by accident. It can repeat things that are commonly known (or more specifically, written about) in the same way that google can surface a desired response to a search query by comparing it against the text people used to link to it (to over simplify things).

You can't tell an LLM to examine its own output and judge it because that's not how it works, that's not how any of this works.