A deep discrepancy between the classical and quantum models of ##L##

[PeterDonis]

Since $[H,L^2]=0$ we have $L^2H|l,m\rangle=HL^2|l,m\rangle=l(l+1)H|l,m\rangle$. So, $H|l,m\rangle$ is an eigenstate of $L^2$ with eigenvalue $l(l+1)$ and so it is linear combination of $|l,m\rangle$ as $H|l,m\rangle=\sum_m \alpha_m |l,m\rangle$. Similarly, since $[H,L_z]=0$ we would have $H|l,m\rangle=\sum_l \beta_l |l,m\rangle$.

All of this is true, but it still does not tell us the specific form of the states $\ket{l, m}$. It's just restating the fact that those states, whatever they are, are eigenstates of $H$ with particular eigenvalues. That's not enough by itself to address the questions you've been asking about the dependence of the states $\ket{l, m}$ on the coordinates. For that we need to know the specific form of those states, which will depend on the specific form of $H$.