Is there any fundamental relationship between the torque on a classical magnetic moment due to a field, and the Lorentz force on a moving charge?

If we have an charge $q$ travelling in a loop of radius $r$ and velocity $v$, then it generates current:

$$I= frac{ev}{2 pi r}$$

We also know that the torque on this loop is given by :

$$tau= I (A times B)$$

Where $A$ is a vector with size proportional to the area of the loop, facing positive perpendicular w.r.t the normal of the loop and the current convention (right hand rule). This is a trivial result the explanation for which can be found many places.

We also define the magnetic moment $mu$ such that $tau = mu times B$

Therefore we can say: $$mu = IA= frac{ev}{2 pi r} pi r^2 = frac{evr}{2} = frac{e}{2m}mvr = gamma L$$

Notice how we are $textbf{not}$ talking about the charge itself but about the loop the charge goes around, if you consider simply that the direction of the electron is perpendicular to the normal vector of the loop it goes around you should see the discrepancy. Think about how magnetic fields cannot do work.