You cannot have a curved threaded rod because the thread pitch must be a constant and the curve makes the pitch vary from small on the inside of the curve to large on the outside of the curve.
Suggest you pivot the entire motor and gear assembly. Looks like you have the hub of the big gear threaded to match the threaded rod.
If so, use a metal threaded rod and match it's thread with the hub of the big gear. Then as the angle of the threaded rod changes, the entire drive assembly will follow it.
With your elegant design, then, then ONLY torque needed is that needed to overcome the friction of the threaded rod/main gear. So, a very small DC motor could easily drive the whole assembly.
Paul