First things first. You know that for the Coulomb potential there are additional invariants, the Laplace-Runge-Lenz vector beyond angular momentum, nonvanishing for elliptical orbits, classically, but vanishing for circular ones.
That is because the "lopsidedness" you are talking about amounts to the ellipse's eccentricity, a dimensionless quantity $|{\mathbf A} /me^2|$, which, in QM, presents as $$ \frac{\hbar \sqrt{2}}{e^2 \sqrt{m}} \sqrt{H(L^2/\hbar^2+1)} \qquad \leadsto \qquad \frac{\hbar \sqrt{2}}{e^2 \sqrt{m}} \sqrt{E( l(l+1)+1)}. $$
You then see this eccentricity analog never vanishes in QM, even for s waves, spherical states! This angular momentum zero-point-shift is well known in deformation quantization, and informs the proper correspondence principle, however counter-intuitive. It has been thoroughly investigated in phase space by Dahl and collaborators.
So, formally, the classical eccentricity survives in QM and is crucial for its spectrum. Recall it is the basis of the original modern quantization of the hydrogen atom, Pauli Jr, W "Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik" Zeitschrift für Physik 36 no. 5 (1926): 336-363.