But again, the amount of energy put into compressing the CO2 will ALWAYS exceed the energy gotten out of it.
Yes, but that is not the sum total of how a heat engine works. You compress the gas, and while it is compressed, you heat it. When it is heated, it acquires more pressure and more volume than that required to compress it initially.
Heat increases volume and pressure. The increased volume and pressure produces work in excess of that required to compress the gas.
Now in order to reuse the gas, you have to cool it back down to it's former volume and temperature, and that requires cooling.
A gas turbine draws in air, compresses the air, applies heat to the compressed air by combustion of the fuel injected, maintains the pressure as a constant as it enters the exhaust(extraction) turbine, and extracts energy from the exhaust gases as they expand and drop pressure. The exhaust turbine and compressor turbine components are connected by a drive shaft; which, feeds power to the air compressor and anything else connected to this shaft, from the exhaust turbine.
The heat energy added to the compressed gas is transformed to mechanical energy within the turbine. If the energy of compression exceeded the energy of the heat addition from fuel, gas turbines wouldn’t work and jet powered aircraft wouldn’t fly.