My confusion is: why would you say that they don't know theory if they know the thing that the theory describes? If a person builds a radio by themselves, would you say that they don't know engineering because they didn't get a formal education on it? If a person figures out how to make a derivative, would you say they don't know mathematics because they didn't study it formally?

Well, yeah... absolutely. The theory describes why it works. I can know "this works" without knowing why it works.