Thank you! I encourage you to see the movie again, absolutely, and check to see if my theory is correct. I was keeping an eye on this during my second viewing, and it seemed that Winter Soldier prefers his right (human) hand for anything that requires fine motor control. The behind-the-scenes explanation is probably as simple as SebStan finding it easier to grab things with his bare hand than the one outfitted in the prosthetic.

Nevertheless, Winter Soldier is *capable* of using his left (metal) hand -- not only is he firing the gun at the end, but at one point he uses it to load grenades into his grenade launcher, suggesting he has at least SOME ability to sense and control the pressure he exerts with it.