Let me crash the party here.
TL;DR: Dispersion interactions are not due to fluctuating induced-dipole attractions, even if everyone tries to explain it that way.
(I know that this must seem like an outrageous statement, and surely downvote fingers are itching now. Bear with me.)
Based on the Hellmann-Feynman theorem, it is known that the forces acting on ...
You are correct that simply stating that the Schrodinger equation is time
dependent does not really answer the question.
In fact, time-dependence is not a component of the derivation of the dispersion forces. At a surface level, they just represent how much the ground state energy of a system of polarizable molecules is lowered by their interactions.