"I suppose that, on a practical level, a take-home for the practicing mathematician is that if you use ChatGPT, don’t trust it to generate valid proofs, and even when it finds a valid proof, don’t be so sure it’s a good proof. And whatever you do, don’t have ChatGPT create a bibliography for you."

In isolation, that's all very good advice. But however is a take-home from what goes before? You used ChatGPT, it did generate a valid proof, it was "a solid by-the-book argument that employed a method I’ve used myself" which may or may not imply "a good proof" but suggests it was at least OK, and nothing in the story you told involves ChatGPT generating bibliographies at all.