The attacker has to be able to issue requests on behalf of the user with injected "canary" strings. I fail to see a practical exploit where one can do this and wouldn't have access to the secret in the response anyway. What am I missing?
Edit: I think I can see a scenario where a third-party website does these requests via an <iframe> or an <img>. I'm not sure there's a way to do POST quite as easily.