What About Electric Fields?

In school we learned about electric fields. Charged particles have electric fields. They are infinite in extent, and the electric field strength falls off as the inverse square of distance away from the particle. We learned that one can calculate the net electric field at some location by doing a vector addition of the electric fields of all the charged particles under consideration. We also learned an operational definition of the net electric field at some location. One takes a small test charge at the point in question and releases it. The instantaneous acceleration of the test charge determines the strength and direction of the net electric field at that location.

Later we are taught that the net electric and magnetic fields are all that matter, that one can have a complete description of electromagnetic phenomena knowing the net fields. In fact, Maxwell's Equations provides just such a supposedly complete theory. We are also told that the underlying fields can't matter since we cannot measure them in any case. Now I would like to take issue with that. Little by little I would like to make the case that the underlying fields do matter and that a deeper understanding of electromagnetism is impossible without including them. The complete failure of 19th-century physicists to arrive at an understanding of the medium of light propagation is a good example.