(This is all assuming the Hvac system is similar to a late w123)

Essentially every part and system in a car can be diagnosed before replacing anything "blind". That is unless you value your time much higher than your money. Though to be fair, testing circuit boards typically requires specialist knowledge and equipment.

I cant imagine a fan will affect the operation of your ACC thermomostatic operation. Essentially that is a feedback loop:
Is the car warmer than I want it to be set? Yes: run the AC some more. No: turn the AC off or drop the cold air output
Is the car cooler than I want it to be set? Yes: turn the AC off or drop the cold air output. No: run the AC some more.

Its a bit more complicated than that, but you get the idea. Even if the fan was not cooling well, it would probably just run the AC more to try to achieve the same temperature.

Before you throw money at it, I would suggest you check each component of the system.

First, does your system get properly cool when it is set to MIN?

Does your AC system produce cold enough air?

Is your heater competing with your AC?

It sounds like you have tested your cabin air temperature sensor already. I would suggest that you pull the switch unit and inspect it. Check for burnt spots on the case, and if you can open the case and check the circuit boards for damage. May be worth reflowing the solder points.

The selector wheel may be removable. If so, take it off and figure out which pins vary in resistance when you roll the wheel. There may be specs available to indicate expected resistance at specific temperature settings. You may be able to take it apart to clean it, do so if possible using some electrical contact cleaner.

Check your grounds. Some sensors measure a voltage drop to ground, if your grounds are bad this can throw off the readings.

Check your fan before you replace it. If the bearings and brushes are good, chances are its fine.

There's probably other checks you can run, but that's what I can think of off-hand.