My point here more than anything else is that I find this formulation unsatisfying because it is "easy" to verify that we have a witness, but is exponential only because the size of the witness is exponential in size.

What I'd like as a minimum example for "give me something that is NP-hard but not NP-complete" Is a problem whose input size is O(N), whose witnesses are O(N), but which requires O(e^N) time to validate that the witness is correct. I don't actually know that this is possible.