Definition of the refractive index

This is not the definition of the refractive index. Its definition is $n^2 = epsilon_r$. The relative dielectric "constant" is in turn defined as $epsilon_r = 1+ chi_e$ where the polarisation $P = epsilon_0 chi_e E$ is related to the applied field by the electric susceptibility. See https://en.wikipedia.org/wiki/Electric_susceptibility.

In the above I have assumed an isotropic medium, at rest, with linear response and relative magnetic permeability $mu=1$. This includes metals. Within these limitations the treatment is generally valid. In the most general case the dielectric "constant" is a rank 2 lLorentz tensor where the elements depend on the frequency and on the direction of $vec k$.

To answer your question, there is no fundamental argument against setting $n=1/sqrt{epsilon_r}$. It is the conventional definition, again with the limitations given.

As it's just a definition, I'm not sure there's any fundamental reason for the choice other than it gives a higher number the more light slows down in a medium and thus the more light is refracted at an interface. One might ask, for instance, why the coefficient of friction was chosen as F/N instead of N/F. Again, larger coefficients mean more friction. So it appears these definitions are useful.