While the stereographic prohection trick sounds neat, it looks like it could produuce a bad triangulation if the input points are distributed non-uniformly. Indeed the 2d Delaunay triangulation algorithm tries to build the most uniform triangulation, but the projection step creates distortion which means a uniform projected triangulation might map to a non-uniform 3d triangulation.

I'm not sure it is an issue because some other factor may compensate this issue, but there should be a rigorous analysis of the stability of this approach to various inputs.