slippery....
The ONLY way to get more energy out of the same amount of fuel is to make less waste heat. Higher compression results in less waste heat transferred to the air by the end of the cycle.
If you confine yourself to framing everything in terms of thermal efficiency you will never be able to explain how or why an engine works.
The topic of this thread is simply "why do turbine engines require a compressor section". That is the question that was being answered by CW. His answer emphasised the requirement for power.
Nobody wants an aircraft that can taxi out on a thimble of fuel before failing miserably to accelerate to flying speed.