Last month I installed a GE 1000-sq ft Window Air Conditioner with Heater (230-Volt; 17600-BTU) in a well insulated and Sheetrocked 750 sq ft workshop/garage. The building is for housing my project cars, power tools, machines, etc., so my goal was to have 40% to 50% relative humidity to prevent atmospheric corrosion.
I was disappointed the unit keeps the space in the low 70 degrees F and about 70% humidity even when the outside humidity is 50%. Measuring temperature and humidity as close as possible to the condensing coil, I found about 45 degrees and 100% humidity. At first I was surprised that the humidity was 100% but then realized that all condensing coils that are dripping wet from making condensation are blowing 100% relative humidity air. As the air disperses and cools the room by absorbing heat, the relative humidity declines. But only to 70% in this case and not to 40-50%.
I complained to GE, and they sent out a serviceman. In a few minutes he said the unit was operating normally and nothing was wrong with it. He went on to say that he also has a workshop with a similar unit and has a dehumidifier because the air conditioner by itself will not make low humidity.
Answering the question "How could an air conditioner cool the air without removing the moisture?", the air conditioner is cooling the air and actually is removing some of the moisture, as evidenced by the condensate in the unit's pan, just not as much as 40-50%, more like 70%. Apparently most all window units are intentionally sized, designed, and priced to only bring down the temperature of a certain sized space and not bring down the humidity to a low level. To get both "room" temperature plus low humidity, units would need to be more robust and more pricey. If using a window unit and you want low humidity too, the solution is to add a dehumidifier which can be had in many cases for around $200 in 2019, not counting any new tariffs. There aren't many dehumidifiers to choose from not made in China and subject to tariffs.