I remember reading somewhere (don't remember where) the many design intricacies of the navigational gyroscopes for the appollo missions.
The gimbals where fitted with sensors and delicate electric motors. The slightest motion would trigger these sensors. The electric motors then moved in such a way that a torque arising from friction was nullified as much as possible. Equally important, the torque motors made the gimbals so agile and responsive that they moved as if they had no inertia
That way the rate of error buildup could be kept sufficiently low to enable the mission.
As you point out in your question, since the angular velocity of the rotor must be sustained the gimbal axes must somehow provide a reliable electric contact, which surely must give a non-negligable friction.
These problems made the navigational gyroscopes very challenging to build.
For the Apollo missions to keep buildup of error under control only 3 gimbals were used, instead of 4. If you use 4 gimbals you can design a configuration that can always be steered clear of a state of gimbal lock
The Appollo mission gyroscopes were susceptible to gimbal lock, when setting up a change of spacecraft orientation the crew had to plan ahead to avoid gimbal lock.
Mechanical gyroscopes are obsolete now for navigational purposes. The successors to mechanical gyroscopse are Sagnac interferometers (also called ring interferometers).
About CMGS
Actually, the purpose of a CMGS is to reorient the space-station as a whole.
That is, the purpose of a CMGS is the same that of a distribution of mini-thrusters for adjusting the attitude of the space-station; the CMGS is for muscling the spacecraft around, it's not a navigational sensor.
The advantage of a CMGS over the array of mini-thrusters is that the thrusters will deplete their supply of propellent, whereas the station's solar panels generate the electric energy that the CMGS flywheels need.
Integration?
That is, if the instrument provides a continuous signal proportional to $\frac{dx}{dt}$, you can find the total change in $x$ of $[t_1,t_2]$ from
$$ \Delta x = \int_{t_1}^{t_2} dt \frac{dx}{dt} \approx \sum_{i=0}^n \Delta t \left. \frac{dx}{dt} \right |_i . $$
Of course, as anyone working on inertial guidance knows, that approximation is makes for real difficulty as does the matter of noise and calibration drifts on the instrument. But if you can sanity check the result from time to time (say by comparing with GPS) this will work quite well.
Best Answer
It's really hard to build an ideal gyroscope. Forces such as friction will tend to cause problems that make the unit precess. Such errors need to be corrected over time if you want to use it as a navigation instrument.
For airplanes, that means that the gyroscopes that drive the artificial horizon are not unconstrained. A simple gyroscope with full range of motion would tend to maintain orientation with respect to the stars. But this unit would also collect errors over time and wouldn't maintain that axis for a long period.
A traditional spinning airplane gyroscope has a mechanism inside that assumes the aircraft is flying level most of the time. The local gravity vector opens vanes that allow air forces to push it slightly. These forces cause the gyro to precess in exactly the direction that causes it to align with the gravity vector.
Modern ring gyroscopes don't erect in the same method, but still require compensation methods for accumulation errors over time.
The video is confusing because when it talks about gyros that are driven in a particular direction by gravity, it shows an image of a mechanism that would not be (the classic gimbaled gyroscope).
If you wanted to do the test mentioned in the video, you'd use a different gyroscope suitable for tracking all axes, not one designed for aircraft navigation with constant corrections and assumptions about local gravity.
Some info on the traditional style gyro can be found on this Aviation SE question: AI in a turn