Otoh, there shouldn't be any movement when he start the engine.

This might be a 2 angle problem.

One, incorrect adjustment of the centering valve. Explains the movement of the steering wheel engine running or not

Two, might be excessive friction in steering joints. For example a frozen tie rod end. Or, my hunch, too much preload on the sector shaft.

A bit of work but, what happens if you drop the pitman arm? Does the steering turn by itself if you start the engine?