In every source I've checked so far, we make the assumption that we can neglect electron-electron interaction in this model but they always fail to give any reason to answer why this is plausible. Does anyone know how physicists convinced themselves of this at the time and even now (other than the fact that it just works of course!)?
The model that Drude originally proposed was a lot crazier than what is now usually meant by "Drude's model". In it, "electrons" were both positive and negative, plus they could carry a multiple of elementary charge e. Oh, and they did not have any mass (but they had an "apparent mass"). He then said that the question why those "electrons" don't get stuck together and form "neutral aether-points" "explains itself" once the kinetic energy of the "electrons" exceeds some certain level. The collisions between them were what determined their mean free path, on which the entire theory hinges.
The answer is known under the name "Landau fermi liquid theory": http://en.wikipedia.org/wiki/Fermi_liquid_theory Basically, electron-electron collisions can only be neglected in a small region around the Fermi surface due. For these electrons, momentum conservation and Pauli principle prohibits scattering. The difficult part is to show that, if electron-electron interactions are taken into account, there is still a Fermi surface.