There is probably something wrong with the pot or the wiring from the pot to the little controller board in the servo.
The simplest test is almost certainly to buy another cheap servo and DON'T TAMPER WITH IT! If that works as expected it's a fair bet that the first servo is broken.
Let's go with the assumption that this servo is broken, because my other servo works fine. Is there anything I can do to fix it, or determine exactly where the issue is?
Your servo appears to not have one.
It appears the pot in your servo is either detached from the servo horn, or the pot has been damaged in some way.
If I take off the metal "V", there will be no stop for the pot, and it should be able to rotate freely. How will this help the pot from locating different positions throughout its rotation?
You can test your pot by sending your servo a 90 deg position command from the serial monitor, then slowly rotate the pot from 0 deg to 180 deg. The motor should change rotation direction as the pot cross the 90 deg position.