Unfortunately I don't think this is enough of a heuristic. I am only speaking about the one language model I have personally used, on character.ai, but it is more than capable of making word play and insightful, often hilarious jokes. Although they are frequently amateurish, I think that's more a function of the fact that I myself am not much of a stand-up comedian, as well as each "bot's" individual training history which is presumably modifying a prompt under the hood and/or training an extension of the model directly based on the conversations.
Of course, in real time the attempts at humor often fall flat and might give away flawed thought processes, although I personally have found them to be often insightful, (containing a seed of humor) even when they're not funny. It could be a useful technique when actually having a conversation, a form of Voight-Kampff test, but I don't think it will do anything to let you know if the content was generated by AI and then just cherry picked by a human.