That above is a pretty common question in fly casting instruction. In my view it depends on your preferences. We tend to consider the wrist as the only joint providing the required rod butt rotation, but all joints act as a hinge so we actually rotate the rod with shoulder, elbow and wrist in varying degrees.
Our torso can translate back and forth, and even rotate around a vertical or horizontal axis. I think that the contribution of our trunk doesn’t get the credit it deserves as it can play an important role in fly casting. In fact, you can cast beautiful loops without using your arm joints at all.
This is a 12 m cast by means of just torso rotation. I am holding the rod with its butt firmly pressed against my belly and not using my shoulder/elbow/wrist at all. As you can see torso’s contribution is not negligible.
One of the greatest things about fly fishing, in comparison with other sporting activities, is that it can be practiced through our whole life; old age not really being a serious obstacle. In fact one of my fishing buddies is 70+ years old now, and it takes some serious effort to follow his rhythm on the river. Paradoxically, aging as a fly fisher is a sort of advantage, as it is the passing of years what makes our experience and technical abilities grow.
I think that, apart from extreme distance presentations, the same is applicable to fly casting, an activity in which you never stop learning and improving. In my view, along this path there are some significant milestones, like steps in the ladder of proficiency. Thinking about it I distinguish four of these “stages of enlightenment”. There are probably more, and you surely will have your own list if you reflect about it; feel free to comment your views.