The motor is a 2-wire device. All it cares about is the voltage across its line leads.
I agree, there's no problem here. Having one input and one output connected is normal (and OK) for an autotransformer, too. I'd look somewhere else for the issue.
Q. Is the transformer sized properly for the motor load, and cabling sized properly for the voltage drop? If either is too small, it would have the same effect as running a long cheap extension cord: low voltage, high current & burnt-out motors.
