robtillaart, I really like your idea of holding diagonally the cube but I'm still thinking in the robot as a stand-alone cube that besides its ability to solve itself, also a human can grab it and scramble it anytime (or even can be used to give the impression that you are solving it!).
Jantje, you idea of using steppers is very good and it will get rid of the gyros but not sure yet.
Boffin1, the RGB LEDs could be an add-in to watch the robot in action in the dark but I see the wiring quite a challenge.
Anyway, here my back of the napkin sketch of the 6 servos. Regards!