Hydrogen storage by physisorption in carbon based materials is hindered by low adsorption energies. In the last decade doping of carbon materials with alkali, earth alkali or other metal atoms was proposed as a means to enhance adsorption energies, and some experiments have shown promising results. We investigate the upper bounds of hydrogen storage capacities of $C_{60}Cs$ clusters grown in ultracold helium nanodroplets by analyzing anomalies in the ion abundance that indicate shell closure of hydrogen adsorption shells. On bare $C_{60}^{+}$, a commensurate phase with 32 $H_2$ molecules was identified in previous experiments. Doping $C_{60}$ with a single cesium atom leads to an increase in relative ion abundance for the first 10 $H_2$ molecules, and the closure of the commensurate phase is shifted from 32 to 42 $H_2$ molecules. Density functional theory calculations indicate that thirteen energetically enhanced adsorption sites exist, where six of them fill the groove between Cs and $C_{60}$ and 7 are located at the cesium atom. We emphasize the large effect of the quantum nature of the hydrogen molecule on the adsorption energies, i.e. the adsorption energies are decreased by around 50% for $(H_2)C_{60}Cs$ and up to 80% for $(H_2)C_{60}$ by harmonic zero-point corrections, which represent an upper bound to corrections for dissociation energies ($D_e$ to $D_0$) by the vibrational ground states. Five normal modes of libration and vibration of $H_2$ physisorbed on the substrate contribute primarily to this large decrease in adsorption energies. A similar effect can be found for H2 physisorbed on benzene and is expected to be found for any other weakly $H_2$-binding substrate.