The problem with circular menus comes down to limited choices (as you pointed out), and the simple fact that it's difficult to read a circular menu. They will work best when the choices are adequately represented by an icon or single word, the choices are hierarchically flat (no drilling through submenus), and when the menu is accessed so frequently that the user can quickly map those choices to muscle memory.* Outside of these conditions, they don't work so well, because their advantages in terms of Fitts' law are greatly outweighed by their lack of readability. Unfortunately a "rings" system would only exacerbate that.

* A circular menu is also especially good if there's some natural mapping between the choices and a circular layout, like choosing a compass direction.