Evaporation is relatively irrelevant by the time you are talking about wind chill. You are not really sweating significantly -we hope. What is really important is the heat of conduction to surrounding air and speed at which heat travels through air.
As Darren G. points out - in still air, your body heats the air around you in a thin layer, which then heats the air around it, and so on… so you lose heat gradually. Remember the old school science experiment where you put a series of globs of wax on a metal rod, heat one end, and see how long before each glob melts? Speed of conduction of heat. Of course in cool air there will be a convection effect too, that heated air around you will rise upward.
A breeze disrupts this, and speeds the conduction of heat to the outside air because the air you heated is pushed away faster, so you are in contact with (and transfer heat to) cooler air. net result you cool faster. Yes, if you sweat you could cool even faster, plus even get a bit colder than ambient temperature. But then, if you took the challenge and dumped a bucket of water over your head, you would cool even faster again. (Partly because wet clothing transmits heat faster than dry “air gap insulation” clothing.) Wind chill assumes a body normal for the situation, AFAIK.
In fact Canada has tried to use some bizarre “Kilopascals per cm^2” measure to better reflect wind chill, but it is meaningless to most people.
A thermometer exposed to the sun would see solar heating effects. Perhaps less if also exposed to a stiff breeze, since the (cooler) airflow would carry away heat above ambient air temperature.
but then, the whole point of measuring temperatures would be to get the average general overall temperature - not to measure sun heating effects or what the local patch of asphalt is doing. local effects and construction can mean significant deviation from ambient air temperature, but generally shade won’t make things cooler than ambient unless the structure providing the shade also absorbs some of the heat (I.e. maybe concrete is cool from overnight or something).
Yes, some conditions can get really bad. When I researched a trip to Egypt, I read someone’s comments about going to Aswan in the summertime. The tarmac felt sticky walking from the plane, until they got to the terminal and realized it was their luggage wheels and sneakers’ soles that were actually melting from the heat of the asphalt. The ambient air temperature may have been a mere 46C but the tropical sun meant that you could fry an egg on the black asphalt.