Currently, once you release the controls, the "AI" brings the turret "back" into "normal" position.

What should be is that the AI should try keep the turret pointed at the same position where it was pointed at the moment the controls were released. Should be possible to maintain even without proper stabilization, as no user input needs to be taken into account - You just keep it pointed towards the same point at all times (or at least make the "AI" try to keep it pointed there).