I don't think that's quite how it works. This device (or any thermoelectric generator) converts a fraction of the heat energy flowing through it into electrical

I don't think that's quite how it works. This device (or any thermoelectric generator) converts a fraction of the heat energy flowing through it into electrical energy. That means energy has to flow through it for it to work - essentially the opposite of insulating. Using this device, by definition, will result in the original hot thing being cooler, and the "cold end" (normally the outside world) being ever so slightly warmer. The place this would be useful is in capturing otherwise unusable energy from low grade heat waste. For example, the output water of a steam turbine might still be "hot", but there's very little you can do with it - it's in the wrong place for heating houses, it's too cold for steam turbines. Generally that energy is lost to the air.