The above explanations are not incorrect, but they don't address the focus of the original question: the comparison of alcohol with water. The questioner was, in effect, saying they knew about "Evaporative Cooling", but had rejected that explanation due to the higher enthalpy value for water verses alcohol.
I believe the answer to this aspect of the question has to due with the quantity of alcohol vs. water that evaporates per unit time. That is, although more energy is removed from the skin per gram of water evaporated, it takes a lot longer for each gram of water to evaporate compared to each gram of alcohol. That is, alcohol evaporates so much faster that it ends up being more efficient at removing heat.
The temperature we "feel" is often deceptive because it's usually a difference we detect, having to do with a lot of other things related to our nerves and their recent history, so there could be other factors related to alcohol's stronger interactions with hydrophobic molecules on our skin (especially oils), or the "drying out" effect that alcohol has after it evaporates from our skin.
But none of this rationalization means that much from a scientific perspective unless actual tests are made. So, I would suggest different solvents be applied to skin and/or suitable skin-like surfaces and the rate of temperature tracked with one of those IR thermometers. Alcohols of different chain lengths, butanol, propanol, ethanol, methanol (warning, poisonous!) would be perfect, but it'd also be interesting to try things like acetone or even oils. The results of those tests would be the real answer.