Given how naturally you can map a grid of quads onto a strip of adjacent triangular faces, I wonder if you might end up with a better distribution of the distortion using a geodesic sphere rather than a puffed-out cube as the basis— at the cost, of course, of an even more hair-raising coordinate scheme for actually addressing it all.

Anyway, you're never going to avoid the existence of some special nodes where three corners come together, and this does nothing to address the altitude problem, but I think it might result in a more uniform surface especially as the overall diameter goes up.