That's a good theory, but by the time the very thin air at 35,000' is compressed enough to keep the cabin pressurized it's temperature is up around 500° and definitely needs to be cooled before introduced into the cabin.

The air conditioning is not the "freon" based vapor-cycle process that we have in our homes and cars. It's an air-cycle process which compresses the air, sends it through a heat exchanger then lets the air expand back to it's pre-compression pressure. This process is repeated at least a second time. The heat exchanger is cooled by channeled outside air.