The energy flux in a conductor paints the picture that the "collisions" of the electrons with atoms (Drude Model) are not the reason for an resistance warming up in the presence of an electric current, but rather through properties of the surface charge distribution. Although of course any magnetic field (in this case) is influenced by the amount of electrons flowing.
Is this the case?
You need more than just the Poynting vector, you need all of Poynting’s theorem$$\frac{du}{dt}+\nabla\cdot \vec S + \vec J\cdot \vec E =0$$
The Poynting vector describes the flow of energy from one location to another. But energy that simply flows through does not produce any heat. In order for the energy to heat the resistor, it must flow in and then dissipate. The term $$\vec J\cdot \vec E$$ describes that dissipation.
Taken together, Poynting’s theorem says that the fields which are produced by the surface charges result in a flux of energy through space, and then the current dissipates that energy from the fields into the matter.
In the Poynting model, the current produces a magnetic field that, in part, guides the energy flow. In the Drude model, the current is composed of electrons, whose collisions with something (lattice defects, phonons, ...) dissipate energy. The potential that drives that dissipation gives rise to an electric field, which produces a Poynting flux toward the place the dissipation is occurring.
