…it has come to my attention that Dick Fischbeck has been working on similar problem. He randomly arranges cones of consistent diameter on the surface of a sphere based on the spherical excess angle of a polyhedron. He has also proposed a variation where the central angles of the cones may consistently generate random tetrahedral pyramids that will tessellate the sphere. Perhaps a careful study of his RanDomes will provide a solution to the Goldberg polyhedra equal central angle problem.
Dick’s RanDome is mentioned in this paper: