What if transform gizmos and shortcuts (G and R) only applied to the 3D cursor when the cursor tool was active?
I know this would break consistency between this tool and other tools but I think it makes enough sense to be easily understood and mostly, it would allow to tranform the 3D cursor with all the snapping and gizmo goodies that come with Blender.
Want to snap the cursor to vertices? No more need to add an empty or to enter Edit Mode and select a vertex and do shift+S > Snap Cursor to Selected. Simply enable snapping and grab the cursor.
Want to move or rotate the cursor along a local or custom axis? Simply select the axis in the dropdown and use the gizmos. Et cetera…
This would be particularly handy for, among other things, transforming objects around a custom and temporary pivot point.