Interesting question.

I don't think the answer is something special to quantum mechanics. Also in the classical Lagrangian/Hamiltonian formalism, conjugate momenta don't depend on their generalized coordinates. So the answer seems to be related to the fact, that a classical state is characterized by two independent values for every degree of freedom.

Time is special, because it is not a conjugate quantity to anything or a function of such quantities. I.e. it's just a parameter and not an observable in Hamiltonian mechanics.