Incidentally - in addition to just the center of mass point, every body / component should just automatically have a «Moments of Inertia» (or similar) group that includes (1) the center of mass as an origin point, (2) any non-degenerate moments of inertia as direction vectors, and a plane when two of the moments of inertia are degenerate (because then any linear combination of those is also an eigenvector so the choice of a particular pair isn't unique)
It should look something like this when turned on:

where the center of mass is a point, the non-degenerate moment(s) are vector direction(s) like parts of an «origin» basis, and the degenerate vectors are a plane (the representation should be translated to center it at the center of mass.
All of those should be usable for defining geometry and relationships (like construction points, axes, and planes).
Just to extend the example of a top, this would be useful because it'd tell you not only where to put the center of rotation, but how to orient it, and how to properly balance it to avoid (or alternately induce) precession / nutation / wobbling / drift.
In an ideal top, the center of rotation would be at the center of mass with the axis of rotation aligned to a large, non-degenerate (unique) moment of inertia, and two smaller / degenerate moments of inertia would be in the plane of rotation