We present a detailed study of the low-energy excitations of two existing finite-size realizations of the planar kagome Heisenberg antiferromagnet on the sphere: the cuboctahedron and the icosidodecahedron. After highlighting a number of special spectral features (such as the presence of low-lying singlets below the first triplet and the existence of localized magnons) we focus on two major issues. The first concerns the nature of the excitations above the plateau phase at 1/3 of the saturation magnetization Ms. Our exact diagonalizations for the s = 1/2 icosidodecahedron reveal that the low-lying plateau states are adiabatically connected to the degenerate collinear "up-up-down" ground states of the Ising point, at the same time being well isolated from higher excitations. A complementary physical picture emerges from the derivation of an effective quantum dimer model which reveals the central role of the topology and the intrinsic spin s. We also give a prediction for the low-energy excitations and thermodynamic properties of the spin s = 5/2 icosidodecahedron Mo72Fe30. In the second part we focus on the low-energy spectra of the s > 1/2 Heisenberg model in view of interpreting the broad inelastic neutron scattering response reported for Mo72Fe30. To this end we demonstrate the simultaneous presence of several broadened low-energy "towers of states" or "rotational bands" which arise from the large discrete spatial degeneracy of the classical ground states, a generic feature of highly frustrated clusters. This semiclassical interpretation is further corroborated by their striking symmetry pattern which is shown, by an independent group theoretical analysis, to be a characteristic fingerprint of the classical coplanar ground states.