You can go back and forth easily between control points and Greville points, using the mathematics of B-spline curves. That makes the Greville points logical choices to use as edit points, because after moving the edit point, the locations of all control points can be computed.
And I guess that because of their averaged nature, the influence of each control point is taken into account at the same level, making it mathematically more stable/robust. The matrix structure is always a banded matrix with bandwidth related to the degree of the curve.