2001 BMW 330 Overheating

Good diagnostic work. Although the auxiliary fan will not be commanded on with the addition of power alone, there are three wires going to the fan, two are for power and the other is a logic wire for bus communication. I would guess there is a problem with the fan, as you have done, but there are two things to deal with here, one is if the fan is inoperative and you purchase a replacement unit and it does the same thing then it leads us to the second thing which could be no logic from the A/C controller to command the fan on. This function can be requested with the factory proprietary software diagnostic unit called the GT-1 and the fan can be commanded on for diagnostic purposes. As a BMW technician, I would start with finding out if the fan is being requested on and if it is and is not coming on, then that should lead me to condemn the fan. The fan and it's controller called the final stage are one complete unit and cannot be purchased separately.