It seems to me that your last paragraph would be a good tactic if -- but only if -- one wanted to find out for oneself whether a particular person is a genuine psychic, assuming that such a person actually exists.
However, if one wanted to publicly verify or expose the person, one would have to have a procedure for proving that the code word is known and verifiable in advance of the reading/performance/whatever.
I suppose the put-it-in-an-envelope-and-mail-it-to-oneself trick would do, but even then someone could cry foul.
Another would be to have the answer in such an envelope, tell the person that you are awaiting a message from the deceased, and if the person gives a message, then reveal the envelope in your pocket and -- either the message is what's written on the paper in the envelope or it isn't.
I think we're back to the thread Robin1 started about about "really real" psychics vs all the fake real psychics.
ETA, where I have written "psychic," feel free to substitute "medium" if you prefer. But why are they always medium? Aren't some of them rather large?