In the previous paper [25], Stolarsky's invariance principle, known for point distributions on the Euclidean spheres [27], has been extended to the real, complex, and quaternionic projective spaces and the octonionic projective plane. Geometric features of these spaces as well as their models in terms of Jordan algebras have been used very essentially in the proof. In the present paper, we give a new pure analytic proof of the extended Stolarsky's invariance principle, relying on the theory of spherical functions on compact symmetric Riemannian manifolds of rank one.