How to solve AC fan induction motor jammed possibly due to magnetic locking/cogging

When normally running from an AC supply, an induction motor runs at synchronous speed minus slip speed. The slip speed is determined by the amount of mechanical load attached. More load and the slip increases to allow the current induced in the rotor to rise and enable more power to be provided to the load.

If you were able to spin the motor and spin it at exactly synchronous speed the current taken by the stator would be zero (apart from the ever-present magnetization current). The voltage induced in the rotor would also be zero.

So we have the scenario that (ignoring magnetization currents), the stator current is zero at synch speed and rises (almost linearly) to some "full value" at maximum mechanical load.

Why should it be a surprise that spinning the motor at faster than synchronous speed (mentioning "sync-speed" absolutely implies it is connected to an AC supply of course) it becomes a generator?

Answering a bit more, the reactive "power" in the rotor does not contribute to output power when driven as a generator. Output power (into say a 3ph supply) is input mechanical power minus generator losses. The magetization currents in rotor and stator are just a means to an end.

A Capacitor Start Motor not only switches out the capacitor when up to speed, but also switches out the starting winding which is in series with the start capacitor. Capacitor Start motors have very high starting torque for a single phase AC motor. The same is true for a capacitor start / capacitor run motor WITH a centrifugal switch. In this case, the run capacitor is a different value and just provides some additional phase lag especially to help the motor maintain speed under load, and improve the power factor. A Capacitor start/run motor without a centrifugal switch has a lower starting torque, but good performance under load, and again, an improved power factor.