It is, of course, of paramount importance that a character being run by the GM be seen as not having an unfair advantage in any way.

The operative words here are "be seen". When you're GMing a game and also playing one of the characters in the adventuring party, you need to be sure to hide all of the little advantages that character has.

Transcript

Pete: An overhead conveyor carries the crucible...
Pete: ... across the factory...
Pete: ... and into the furnace room.
Padmé: Oh that can't be good...
Pete: Of course, R2-D2 flies to the rescue.
Pete: There's no question of R2-D2's loyalty to his fellow PCs.
Pete: Despite the situation being hopeless and Padmé facing certain and horrible death, R2-D2 does not abandon his compatriots.
Padmé: Wait, tell me if I'm completely off-base here... but am I detecting a hint of reproach?
Anakin: Is this the entire reason this factory is so dangerous?
Pete: Dangerous? R2 is having no problem. It's child's play!
Anakin: Right...
Pete: You'll note that I've constrained R2's actions using a behavioural simulator on my PDA, to avoid any accusations of bias.