What is the essence of electron-phonon induced Charge Density Wave (Pierls Instability)?

In the context of electron-phonon coupling induced Charge Density Wave (CDW), let's use the simplest 1-d metallic chain model as an example, where there is Peierls instability, the most common approach to see that is based on a mean-field kind of treatment of the lattice distortion ( see Gruner_Density Waves in Solids ), which is, from my understanding, if we assume a small position deviation of the atoms, which will lead to a gap opening at the original Fermi surface in the electron band structure ($Q_{cdw}=2k_{F}$, and the electronic energy get lowered ), then calculate the total energy of the system ($E=E_{el}+E_{latt}$) and try to minimize it, we would finally get a non-perturbative formula for the order parameter. If one also try to calculate the charge density of the ground state, one would get something like:

$\rho(r)=\rho_{0}+\Delta \rho \$cos$(2 k_{F} \cdot r+\phi )$

as a result, people say that this charge modulation might carry the current when add external electric field. (Frohlich_On the Theory of Superconductivity: The One-Dimensional Case)

My question is, what's the difference between the charge density wave phase and a normal insulator phase? It seems that it is just a matter of changing the periodicity of the lattice ( or the periodic potential felt by electrons ), at least from the mean-field treatment I think that's the point. So why cannot one think in terms of electrons moving in a new periodic potential, which finally gives a insulating band structure, and which is obviously an insulator? I know this is wrong but at least currently I couldn't get the reason. Hope someone can help me with this problem.