So after going through a half dozen motors, we've taken the plunge and actually removed the back off a (seemingly) working motor to find out whether we can make a legal repair to our motors. This resulted in the following:
The closer inspection of the part showed that even under seemingly normal loads we're pushing the limits, as seen from the burning marks by the white band and the crack that stretches from one end to midway across the backside.
It seems like the inductor we pulled out is a 1.9uH inductor, with a resistance of 0.3 ohms. Tomorrow we're hoping that some tests with an o-scope will help with verifying the inductance value and get to the bottom of this mystery.