Let's start with this one:
The other thing is that I don't beliewe that you can extract mechanical power from hot air. Car's supercharger (turbine and compresor stage on the same shaft) doesn't use temperature but pressure - meaning it doesn't lower EGT but it only ads backpressure.
Any turbine uses
temperature and pressure together to extract energy from an air or gas stream. Can't have one without the other; it's the natural
adiabatic property of air. A jet engine with multiple turbine stages may drop the gas temperature 200-300C per stage, at least in the HP turbine.
This basic thermodynamic process is the big part of how an air cycle machine works, as well as any other turbine.