Platinum is an excellent catalyst for H+ reduction to H2 and H2 oxidation to H+. Therefore the SHE electrode is based off of a platinum electrode for that reaction. Platinum is robust (stable) and highly active, which is why it is used in that electrode.
The reaction of a reference electrode should be reversible with low overvoltages. Furthermore the reaction should not be limited in order to have a low impedance reference.
The best system according to that is Hydrogen at Platinum.
Any other reaction is slower or less reversible.
Furthermore the reference system should not poison your electrodes. Any other reference is based on Hg, Ag, Cu, Cl, SO4 etc. the reference may poison your system.