It sure looks like that thing has a thermostatic control. I’ve got similar window shaker units and they have a distinct click when the thermostat turns on or off and they do an ok job of controlling the temp in the bedrooms.
OP here. Wow, that's a good point. I don't know why I assumed that the first knob is just a static control for cooling output. Especially considered that it's labeled "Temperature."
Looking at some customer reviews, I do see some complaints that it doesn't work very well on this model. But still, I should have read the manual. Thanks for correcting me!