How did you prompt it to play tic-tac-toe? I'm surprised that didn't work, it feels like something it should be able to handle really well.

Hangman and stone-paper-scissors though are entirely unsuited to a language model, at least one with a chat interface like ChatGPT, because they both require it to be able to store a secret. ChatGPT has no ability to do this: each time it returns a response by evaluating the previous conversation.

You could build a system that COULD play those games via an LLM but you'd have to write extra code to do it.