I know of no canon answer, but I would guess that it has more to do with mental focus; for example, when force choking someone, he's basically using Telekinesis crush the target's throat. For the purposes of focusing it, it's probably easiest to visualize a hand doing the task. Then, as it crushes, it's not simply compressing, but rather simulating the action of fingers bending inward; using his own hand, both being controlled (albeit by servo-moters) and visible in front of him helps him properly visualize what he's doing.
The same idea applies to shoving motions when doing a Force Push, and grasping motions to summon a light saber; in each case, what he is doing with the force is a mimicry of what his body is doing. Given that, even via prosthetic, the ability to do the tasks physically is something long since learned, I think he's doing it physically and using the force to extend the reach via what amounts to simply a remote, invisible, force based hand.
Could he do it with no gesture? Sure; but using a physical gesture as a template removes the need for extra thought to makes sure he's doing it properly. (If he's going to choke someone by imagining a loop around their neck and tightening it, he has to consider where the cord comes together, is it simply constricting, or being pulled tight, etc. But imagining his hand doing it requires no thought.. he's done it physically many times.)
Another aspect may be the intentional limitation of force use. Much like the question Why does Q click his fingers, I think Jedi develop personal gestures or use mimicry to focus their powers to limit themselves; without some 'trigger' for the powers, what accidents could be caused by stray thoughts, for a strong force user? (A passing women inspires a dirty thought? The passing lady gets goosed. Feel a sudden urge to shove someone out the window? It happens.) A minor version of an action to be performed on a larger scale WOULD tend to give your actions away slightly, but would make it much harder to accidentally perform a force based action.