Other people have already mentioned the distinction between turbojets (what you’re describing) and turbofan engines, but I think there’s another inaccuracy:
> They produce very little power, just enough to power the Auxiliary Power Unit (that manages the plane electronics, air-conditioning etc.).
The APU is a completely separate gas turbine that doesn’t rely on the main engines. As a consequence, the APU on an airplane will also have its own exhaust.
> If you want to use a gas turbine for producing power, you will set it up such that most of the energy goes into the work generated, rather than the exhaust, so it would be a cooler, slower exhaust, similar to an IC-engine.
Yes, obviously. At the same time, I was under the impression that turboprops and turboshafts on airplanes and helicopters still produced enough jet exhaust to represent a safety hazard, and in those applications you would also expect that most of the energy would go into the work generated rather than the exhaust. So is it just that this residential generator is even more efficient than the main engine of a heavy-lift helicopter? Is it because it’s a less powerful machine in the first place? I could continue to speculate about this but I don’t actually know.