In real world AC applications, if the outside air is colder (or substantially colder), then you run pumps, not compressors, and push the fluid around to move heat energy from inside to the cold outside. You could do this with water or glycol instead, but refrigerant is way more efficient. You could also just push air between the outside and inside, but then you have humidity issues and potentially pollution/contamination issues, which a refrigerant exchange avoids. If the outside air is warmer than the inside air, you switch to compressors and use the phase change to both overcome to inverse heat gradient (which you could not do at all without some amount of compression), and also because the energy exchanged in phase change is super efficient.

While on paper you might be able to keep everything a gas but achieve some comparably insane pressure to get from say below 50F to WAY above 150F on the inside and outside respectively, it's way easier to use a gas/liquid phase change.

You also have other benefits with gas/liquid... you can have an accumulator or a receiver that holds liquid and effectively reduces the refrigerant charge if there is too much refrigerant for the current operating conditions... and you can flood the refrigerant back into the outdoor condenser to effectively reduce its capacity if you are operating in very cold outdoor conditions but still need DX without freezing your inside coil (or don't have the option for pumped refrigerant economizers).