Well, it could work. What you are doing is exactly what an air conditioner does. It cools the air and water (in this case, liquid heat) falls out of it.

In your house, you get nice cool dry air and a stream of water draining out of your a/c system. They are basically proposing doing the same thing on a massive scale.

They want to air condition the planet!

Which, if resources are no problem and energy is limitless, can be done. However....

It's a palliative not a cure. Pulling water vapor out of the atmosphere could cool down the planet but the CO2 is still there for the next 4-7 thousand years. That won't change, and as soon as you stop that CO2 will start driving up the temperature again.

If you are going to spend that much effort on a global project like this. CO2 removal from the atmosphere is the only option. Every ton of CO2 pulled out of the atmosphere is gone forever (if we do it right) and that will cool the planet down.

Even if civilization collapses and the project grinds to a halt, every bit of CO2 removed will have helped make the planet cooler. It's the better way to go.