400% efficiency is good, but it’s not better than the ∞% efficiency you get from something that doesn’t require fuel input to begin with. (I’m pretty sure the Technology Connections guy would agree on that point.)
If nothing else, think of it this way: even if you still want to use air conditioning to make sure you get all the way down to comfortable room temperature or whatever your target is (which a Qanat, although able to achieve a >15°C ΔT, might or might not be able to do reliably), it’ll still give you a big head start and greatly reduce the amount of energy needed. It’s a lot like using a ground-source heat pump instead of an air-source one. What’s not to like‽