FADEC has nothing to do with flight controls.
Total loss of electrical power should not result in the engines shutting down (absent some other defect like the fuel suction pumps not working).
The consensus of publicly-expressed ostensibly-expert opinion is that there is no asymmetric power-related yaw after take-off but, even if there were due to one engine not delivering 'sufficient' thrust, the other engine should have been capable of delivering enough thrust for the aircraft to climb away.
The consensus of publicly-expressed ostensibly-expert opinion is that the bang and revving heard by the survivor was likely the RAT deploying and revving up.
Everything points to the immediate cessation of motion lotion to both engines, shortly after take-off. If that happened, the question of course resolves to: Why?
And a PS to my earlier post about the schematic in the TCMA patent application: Apparently the TCMA can be triggered even when the thrust lever is not set to idle. There's an envelope of engine thrust delivered compared with thrust lever position, outside of which envelope - 'too much' thrust delivered compared with thrust lever position - the TCMA is designed to trigger fuel shut off to the engine to which it is fitted, provided the aircraft is not in the air. There remains uncertainty as to how TCMA decides whether the aircraft is not in the air. My understanding is that there's a combination of sensor inputs, like the obvious weight on/off wheels as well as RADALT, but I can't find any authoritative statement of what those are and whether just one or both (or all have) to be in the 'in air' or 'not in the air' state to disable or enable the TCMA function.