Hi - I'm afraid that the problem might actually be the MOSFETs. Long story short, you should be able to fix the problem by buying different MOSFETs (I will give links at the end), but here is the complete explanation.
This gets a little deeper into electrical engineering, but here is the datasheet for that part:
https://www.infineon.com/dgdl/Infineon- ... 3b3a9f220d
The relevant spec is VGS(th) Gate Threshold Voltage on the second page, which has a *range* of 2-4V, meaning there is some variability in the manufacturing and it will vary from part to part.
I won't repeat it all here, but to understand how the circuit works, you need to read the FAQ starting with the "how does the light sensor work" question:
https://www.sciencebuddies.org/science- ... #question6
When you remove the potentiometers from the circuit completely, you are removing R2 and R4 from the circuit diagram in Figure 4. We'll just talk about the left half of the circuit since they're identical. When you remove R2, that means the voltage V2 (which is the voltage that turns the MOSFET "on") is *always* going to be equal to the battery voltage, which is the full 3V. Since you said all the robot's motors work in this scenario, that means 3V is enough to activate all of your MOSFETs.
However, when you insert the potentiometer, the voltage V2 is going to be "pulled" even lower. If you turn the potentiometer all the way in one direction such that the resistance is zero, then V2 will go all the way down to 0V, and the MOSFET will be fully off. If you turn the potentiometer all the way in the other direction, that is going to push V2 higher. However, depending on the maximum resistance of the potentiometer and the gate threshold voltage of the MOSFET, it might not be high *enough* to turn your MOSFET on, which is what I believe is happening for 13/15 of your robots. So, the potentiometers are not broken - they are working exactly as they should, but their resistance is too low, and your MOSFET's gate threshold voltage is too high.
Where does that leave you? Here's what I'd recommend:
1. If you haven't already tried this, you should be able to confirm my theory by taking the MOSFETs out of one of the two working robots, and swapping them into one of the robots that currently isn't working. If that robot suddenly works, that confirms my theory that the issue is the manufacturing variability in the gate threshold voltage of the MOSFETs.
2. If you want to get all the robots working, you need to buy more MOSFETs with a lower gate threshold voltage. I have had good luck with these:
https://www.amazon.com/gp/product/B07CTF1JVD/. Note how on the datasheet, the gate threshold voltage is lower (1 to 2V):
https://www.sparkfun.com/datasheets/Com ... 0N06LE.pdf
3. Alternatively, you could buy potentiometers with a much higher resistance value. Assuming you bought 10K potentiometers, you could go up to 100K or even 1M. But I do not think that will be as reliable as option #2.
I'm not sure how much, if any of that will make sense - if it doesn't, don't worry, because this is really college level electrical engineering. Again the short version is that there is nothing wrong with the potentiometers; the circuit was designed to work with a specific type of MOSFET and you wouldn't know that without looking at the datasheets.