Fan Capacitors are usually only a few uF
2.2 uF and rated for 400 V dc or 250 V rms ac.
A cooling fan has low torque. Often the Capacitor remains switched on as the currents are low.
The transformer / inductor is in series, often all the time and causes a bigger phase shift and reduced voltage on the motor windings, hence more slip and different speeds.

The product of rotation, excitation and flux produces electricty.