On the cold side of a turbocharger, which is pushing air into the engine, it is more efficient when the air is cold. Intercoolers are even used to cool the air. So the heat from the exhaust seems to be a problem, rather than a benefit.
Edit: Just adding that the heat in turbos seem to be less of a problem than it used to be. More advanced materials perhaps? Definitely no turbo-timers anymore!
The cold side heats up due to the compressive 'forces' (or whatever it's called) AFAICT. Same reason superchargers need also need aftercoolers to run at peak efficiency.
One of those pesky laws of thermodynamics seems to imply that if a turbo only used the kinetic energy of the exhaust it would not be able to overcome the losses of transferring the energy into the intake and would be a net drain on power.
Of course, I'm no thermodynamic engineer and am more likely to be wrong than not.
--edit--
Though, certainly, a turbo is harnessing some of the lost energy of the combustion process since the piston can't do a 100% energy conversion.
The reason turbos work is not because they are forcing more air, it’s because they are forcing more air which ALSO raises the compression ratio and the amount of fuel detonating (the MAF will sense the increased air density and add additional fuel to compensate). I.E. a turbo burns more fuel and therefore it produces more power.
Hot air is less dense than cold air (why heat rises). Probably keeping the hot side of the turbo hot helps with gas expansion, perhaps that increases the force on the turbine? I’m not sure about that. But the intercooler is to cool the air on the compression side because cold air is denser. Which means more air into the cylinder, which means more fuel. Probably the colder air helps with auto detonation too which lets you push the compression ratio further if the pistons and cylinders can handle it.
Turbos aren't really about transferring energy into the intake. It's about getting more air (oxygen) into the intake, so you can burn more fuel per engine stroke. The extra power comes from combusting more inputs.
A gas heats up when it is compressed.
This innovation was key to the success of the V2's rocket engine.