Yes Electronic diff lock is a variation of ABS or TCS. It is entirely seperate from the gearbox. It is very similar to traction control except traction control applies both front brakes and reduces throttle, electronic diff lock applies brakes only to the wheel that is slipping. Audi definition is given below:
Electronic Differential Lock (EDL) applies brakes at driven wheels as necessary
As I said this brakes the wheels, not the gearbox. The fundamental principles of a differential, which is part of the gearbox on a front-wheel drive car, means that if one driven wheel spins, all the torque is diverted to that wheel, hence by braking the wheel that is slipping (the EDL) torque is transferred back to the wheel that is not slipping, and hey presto hopefully forward motion will be restored.
If it braked the gearbox the gearbox would need a limited slip differential, and I assure you it does not have one. On Volkswagen/Audi models the only vehicles that have a limited slip device are the AWD models, with Haldex coupling for Golf, A3 etc, and Torsen LSD for A4, A6, A8 etc.
You can't have a failsafe mode on the gearbox where the diff is locked open, because the diff on nearly all cars is open anyway, unless you have a limited slip diff, which 99% of cars don't.
In your case it sounds as if the differential itself was damaged in someway so that power is not getting to the wheels, but this sounds like a very serious manufacturing defect and you should get compensation as well as the gearbox replaced under warranty.