It works in less arid places too. You can use the humidified cool air to cool incoming air. The now warm air is pumped out. Essentially, you can achieve this by putting a wet filter on the house outflow of an MVHR system, before the heat exchanger. The problem with MVHR is that the flow rates aren't very high. I have an inkling though that with a well designed, efficient house you could maintain a more consistently comfortable temperature with very low energy usage (think a few W and perhaps 1 l/hr water). The psychrometric chart suggests you can get a temperature delta of better than 10 degrees Celsius at 35 degrees with 40% RH.
I don't think so, though I did struggle slightly to understand the schematics. That system still seems to humidify the incoming air, just in two stages. I might have missed something though.
The point is if we don't care about losing the water to the atmosphere (in climates where water is not scarce), we don't need a closed water cycle. For your house outflow air, on the psychrometric chart, you follow the isenthalpic line with evaporative cooling as far as though can (the diagonal lines from bottom right to top left ending in the wet bulb temperature, which is adiabatic in our case since constant pressure). This gives the temperature on the cold input side of the heat exchanger, which cools the incoming air, which will have an increase in RH, but only due to temperature change.
I posit the system would work well to reduce uncomfortable temperatures to be more in normal range in temperate climate extremes.