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I expect you're either both right or wrong, depending on how you want to look at it.
Forget quibbling over the filter, the air, the fan, whatever. Just consider yourself in a closed system.
Evaporation is an endothermic reaction. Energy is "used" as part of the state change. This energy comes from the surrounding environment, but the temperature of the water does not change during evaporation.
The ambient energy expended reduces the heat in the environment. Less heat in the same materials will result in a lower temperature, which is to say that evaporative cooling is real. So the Mrs. is correct.
Does THIS device provide enough to actually meaningfully cool your space? Tough to tell. You could weigh how much water you're evaporating, look up how much energy that expended
https://www.engineeringtoolbox.com/water-properties-d_1573.html
And then try and rough out how that would translate to a cooler room with specif heat capacity of air....
But honestly I'd probably just try and ignore all the interactions and just use a thermometer at the output of that thing to see if it's at all different from the ambient temperature of the room.