Without the pressure differential the heated air would still do work, it's just that it wouldn't be much use. Gas flows from a high pressure region to a low pressure region. No compressor and the heated gas, which is now at higher pressure (pv=nkt) sees lower and equal pressure at either end of the tube. Compressor (or ram effect, or reflected shock as in those exhaust pipe trick engines) gives the right gradient to send the gas to the designated exhaust where it expands, cools, and does work. The higher the initial pressure of the incoming gas, the higher its density and so it can support a richer mix, which increases temperature,etc.