From the perspective of an electronics technician, following are several notes to reply to your message. There is no straight-forward answer here. So I will try and keep the techno-jargon to a minimum, unfortunately it will not be a short message.
You are correct that a potentiometer is a varible resistor and is (usually) a 3-terminal device. The total resistance of the potentiometer is between the two outside terminals (I.E. # 1 and 3) and the middle terminal is the variable wiper (I.E. # 2).
If the potentiometers resistance is 100Kohm (K = 000, therefore 100K = 100,000-ohms) and you measure with a volt-ohm meter (for resistance) across pins 1 & 3, you should measure around 100,000-ohms (+/- tolerance of potentiometer). However, if you measure between pins 1 & 2, and adjust the potentiometer, you will see that from one end of the scale to the opposite end, the resistance will change accordingly (0 to 100,000 or vice-versa).
If you have a potentiometer with more than 3-terminals, it could be a stacked or twin layered device for a stereo volume or balance control. If you have a terminal on the opposite side of the potentiometer from the 3-terminals, this is usually a mid-span "tap" used in audio applications (no need to cover here). If you have terminals on the back surface, these could be the on/off switch inside the potentiometer.
BTW - a potentiometer in a seteo amplifier works quite differently than in a PSU fan control circuitry. In a stereo, at the low volume settting, the resistance is zero (zero volts of audio) and at the high volume setting, the resistance in maximum (higher audio voltages). his can easily be seen with an aid of an oscillscope.
Enough about potentiometers. Let's continue with your modifications....
When you cut the thermistor from the circuit, the fan control circuit was basically turned off and when you connected the wires together, the fan would receive full voltage from the control circuit. Good this tells me the control circuit & fan still works.
BTW the fan control circuit is "normally" a dual-stage circuit. In other words, a power source, 2 transistors, a few resistors, the thermistor and the fan.
Thermistors usually have a negative temperature coefficient. In other words, the cold resistance is more than the hot resistance. The temperature around the thermistor varies the resistance in the thermistor, thereby affecting the current through the device. The thermistor controls the fan circuitry and when the thermistor is hot, the fan turns faster - thermistor is cold, the fan slows down. Simple really.
When you installed the potentiometer, the fan started to work. At one end of the scale, the potentiometer provided "zero" resistance and the fan received zero voltage and the fan would be off. At the opposite end of the potentiometer scale, the resistance will be low and the fan will make the "Concorde-mode" racket. This potentiometer needs to mimick the thermistor (negative temperature coefficient).
I would have predicted the results as you had received with the potentiometer. BTW the picture provided is correct for installation of the potentiometer. Now comes the tricky stuff. You said you had placed the thermistor in series with the potentiometer and things got weird. Remember the negative temperature coefficient? That is going to throw a wrench into your circuit.
With the potentiometer at the lowest setting (zero resistance) the thermistor will be the only resistance in the circuit. Then when the thermistor is hot, the thermistor resistance drops and the fan should run faster (same as it had originally before this mod). However, if the potentimeter is set to the opposite end (high resistance) and the thermistor heats up, I doubt whether the fan will even start up.
What I would have done is different. Lets assume the thermistor is 100,000-ohms. I would have chosen a potentiometer around 500,000 to 1M-ohms (1,000,000-ohms) and intall in in parallel to the thermistor.
Your statement that resistors in series are additive is correct. But in a parallel configuration, the resistance can be cut in half (assuming equal resistances). In the case of a 100K-ohm thermistor in parallel with a 500 to 1M-ohm potentiometer, the lowest resistance will be slightly lower than 100K-ohm. This should not pose a problem here.
At one end of the potentiometer scale, the fan would be driven hard (on) and the thermistor would have no effect. At the opposite end of the potentiomenter, the potentiometer is mostly out of the circuit and the thermistor would be the automatic control function. To adjust for better fan control at rated temperatures, you would need to play with the potentiometer to find that happy medium.
At this point, further examination and evaluation of the fan / thermistor / potentiometer control circuitry needs to be done. It is difficult for me to provide an end-all, beat-all idea without first trying it.
Did I answer your questions, of just added more mud?
Good luck and happy compting.