Not following you. Do you agree with the following ? If so do you agree the fan provides cooling ?
When water evaporates it consumes heat and carries it off. The amount of heat consumed is well known. "Values for latent heat of evaporation (λ) of human sweat have been debated in the literature, considering the effects of temperature, humidity, and sweat osmolality, but suggested values ranging from 2,696 J/g (13) to 2,595 (27, 31) and 2,398 J/g (29) have finally converged to the latent heat of evaporation of pure water (33), only dependent on temperature giving a number of 2,430 J/g at 30°C".
The fan comes into play by changing the amount of sweat that evaporates on you cooling you vs the amount of sweat that falls off you and hits the floor. I believe the fan does this by changing the local water vapor density next to your skin. More air flow, more evaporation, more cooling. Same reason a wet bulb thermometer shows a lower temp under higher winds. Same for wind chill calculations.
There is also a convection cooling effect from the fan's airflow. Your skin temp is around 98F-100F. No air movement, the air around your body sits at the same temp, 98F-100F (ignoring some relatively slow convection currents). With fan induced airflow, larger amounts of cooler air comes in contact with your skin increasing the rate you can dump heat. There is a convection cooling effect even if you are not sweating if ambient air is less than 98F-100F. Think windy day in winter vs calm day.