Thinking about this a bit more, I think we do have to count phase changes differently. The key is that the movement of air around an evaporating liquid (or sublimating solid, etc) is not removing heat by itself heating up and thus providing a conveyor of heat. Indeed the moving air can actually be hotter than the evaporating liquid, but still acts to cool it down.
The point is that the moving air is reducing the concentration of free molecules of the liquid at the boundary, and thus moving the equilibrium so that evaporation proceeds. The air need not even be moving, just diffusion of the evaporating molecules away, or even in a vacuum, the molecules leave the surface, and the net effect is that the partial pressure is reduced below that of the liquid, allowing the phase change and lowering the temperature.
A sweaty human is cooled by a moving wind, even if that wind is hotter than the sweaty human. This is intrinsically different to convection or conduction, both of which require the air to be cooler than the sweaty human.