The aux fan theory is interesting and worth checking out. However, in my experience it should still cool OK as long as the car is moving (and thus you get ram air over the condensor). On the other hand, if it doesn't cool when in stop-and-go traffic but then cools off nicely once you start moving again, that would point to a possible aux fan problem.
Another simple idea. When cabin temp is a lot higher than set point, like when your car is sitting closed up in 100 degree heat, the system will call for recirculated air. Obviously, if the air in your car is 120 degrees to start with, even if your AC, aux fans, etc. are working perfectly, the output air will still only be about 30 degrees cooler (in this case, 90 degrees) and thus feel warm.
Does it help if you open all the windows for the first mile or two to get rid of the superheated air inside?