Why would a rotor get stuck on the side of the stator when the induction motor is turned on, and how to fix it?
With a capacitor problem, the motor would not start, but it wouldn't "get magnetically stuck." The motor would start when given a "push" by turning the motor or fan by hand when the power is on. That should be done carefully, perhaps using a stick. It should only be attempted with a low power motor.

It is more likely that a bearing is so worn that it is not holding the rotor in position properly. There could also be a problem with the motor housing.