In its simplest form, I think a thermocouple is just two wires with different electrical conductivity connected to each other in a hot place, with their ends sticking out in a cold place, right? And, you’ll get a voltage across the ends in the cold place, and that can be used to do work.
So, take those two wires and don’t connect them to each other, but still have one side in the hot place and the other side in the cold place.
It’s not clear to me, that the coffee will cool down more slowly when the wires are disconnected than when they are connected.
But, wait! The wires are doing work! Yes, but in my (probably inapt) analogy of the spinner connected to the bottom of the cup, the cup will drain more slowly when the spinner is doing work than when it isn’t. Using the potential kinetic energy of the coffee in the cup while draining it slows down the drain effect. Why does using the potential heat energy of the coffee cup necessarily speed up the use of that potential energy?
And, thanks for that first answer, of course, but I’m hoping to get a definitive answer.