No, those motors are only powered when the mercury switch detects the gyro is out of level, and places a very low torque on the axis to restore the gyro to the central level. It is very low and slow, taking minutes to do so so that the motion of the aircraft does not cause it to show an incorrect attitude. It averages out the switch operations so that only the nett effect of a long period is acted on. The motor is a split capacitor motor ( thus the capacitor on the gyro frame in the video) with the switch selecting a direction for reaction, or nothing when the axis is close to level.The fast erection switch simply shorts out a series resistor to change the action from minutes to seconds. If you are not flying level the fast erection will lock in an incorrect attitude that will take a long time to level out.
You do have synchro outputs from gyros, but these will not have a motor style core, but will only have a single pole rotor, fed via slip rings, and a 3 phase stator winding. This is normally used to drive either a computer or a 3 axis display ( a ball) remote from the gyro, allowing the gyro to be placed at the mass centre and the display to be at the cockpit a way away, where it will not be affected by false readings during turns or pitching.