This is basically a technique that's been used for possibly millenia in Persian deserts. It doesn't require great amounts in many configurations. In fact the drier the surrounding air the more effective it is.
Dry air makes the water evaporate more quickly, but it doesnt cause the water to carry away more heat when it does so. Per unit heat, you will use as much water in dry air as in air with some moisture. You will get more cooling out of a smaller area/device, but wont use less water to do so.
And if that were all they were doing you'd be right. But it's not. High humidity impedes radiative cooling (they use both evaporative cooling as well as radiative cooling). Low humidity allows their design to cool more efficiently then it would in a high humidity environment.
These things are extremely well studied by people who know what they're doing. I'll never not be impressed by the ability of a random on an internet forum to 'well akshully'.
But the radiative cooling is going to happen with or without the water. It is a different system operating within the same device. Routing air underground to cool it will work regardless of water, at least until things heat up underground, which can be a thing in low-conductivity soils.