How does an ultrasonic humidifier / atomizer affect temperature?

Evaporative cooling is a technique used to cool air by using the evaporation of a liquid.

My google research on evaporative cooling shows a ubiquitous concept of forcing air through a pad/membrane/foam/sponge that holds water, and the air exiting the pad is colder to a degree that depends on the ambient humidity. Apparently the output can be 10C or more colder than the input.

(Image from wikipedia.org)

As I understand, it’s the evaporation of the water that causes the air to be cooled. And, as I take it, it’s the heat from air at the input being absorbed by the water and causing a phase change that cools it.

What if the water is atomized by something such as an ultrasonic humidifier?
Ultrasonic humidifiers use a “speaker” of sorts to vibrate water into a vapor. I can’t find any mention of ultrasonic humidifiers being used to cool a room.
At first it seemed to me that they would not cool the air, as water was simply atomized (instead of the heat from the air going in to a phase change).
But thinking about this another way: The atomized water particles would mostly evaporate in the air, which should thus cool it, right?

Do ultrasonic humidifiers cool air to the same degree that pad-based evaporative coolers do? (Assuming similar water volume used). If not, can ultrasonic humidifiers be used in a more elaborate system to cool the air?