Would it be accurate to simple define what a hamilitonian operator is as "The only operator that commutes with every other operator"? Basically, given a system with some symmetries, the hamiltonian is the only observable whos measurements wont be affected by any symmetry transformation, e.g rotations wont change the expected values, and neither do translations (if thats a symmetry) and idem for any weird symmetry you impose upon a toy system. So the most symmetric of all observables=literal definition of a hamiltonian.