But it isn't that simple. The insight is in the importance of using the centroid of the shape and the path it traces. It can't be any other point for the simpler form to work.

That's the cleverness, and the reason the centroid has that property is interesting enough, and not immediately obvious. But by understanding why it works, we can see we can apply it for much more general cases than surfaces of revolution, which is usually the only treatment the theorem gets out there. The purpose of the post was to illustrate the generality of the theorem.