Servo steering with 88004 motor and 8878 battery box? I would like to build a servo steering (not remote controlled). Besides the direct physical connection between the steering wheel and the wheels, the rotation of the steering wheel would be used to control the auxiliary 8878 motor through the 88004 battery. Is it possible to tune it properly to achieve a precise servo steering? Since I don't have the battery box, I cannot do any experiment yet.

The schematic layout would be something like this:
There is a bit of confusion in your question with the part numbers and their location on the diagram. That is, you seem to have them reversed.  Unfortunately I did not have a rechargeable version of the "AAA" battery box to test and prove the concept, but I was fairly certain it would work based on what I know about these battery boxes.  Having a 7-speed variation, in each direction, makes them function very similar to the "Speed Remote" (part# 8879). I can confirm that the Servo Motor (part# 88004) will turn incrementally using this remote.  It therefore seemed reasonable to assume that the Servo Motor, connected directly to a Rechargeable Battery Box (part# 8878), would function in the same manner.  Confident this would work, I sought out and found a couple videos on Youtube that proved this to be true.  

The video by Lasse Deleuran: https://www.youtube.com/watch?v=tzQ-B7xfJjQ
The video by Sariel's LEGO® Workshop: https://www.youtube.com/watch?v=T_Gvuzsngg8  

So the answer to your question is "kind of", depending on how precise you need, or want, it to be.  The servo moves in 7 increments in each direction, which is a great improvement over a standard battery box or controller.  For your build you would simply "input" the steering wheel shaft into the orange control switch of the Battery Box.  It's position and steering wheel-to-input ratio could be manipulated with gearing if desired.  


(screen cap from Sariel's video)  

An updated verison of your diagram would look something like this:  

  

The servo motor has axle holes on both sides of the output.  In theory a steering shaft could be ran into the rear of the output to provide the redundancy you're looking for.  Some amount of "play" in the physical connection would likely be needed to allow the wheel to turn just enough to activate the switch.  Because of the nature of the 7 position servo motor, choppy steering is unavoidable.  Full-electric drive-by-wire steering, while not used in passenger vehicles, is very much a "real system".  Not knowing the purpose or application of what you're trying to achieve, it's all just speculation.