Energy Transport in Circuits
This seems to suggest that you would physically need this number of electrons to pass between the terminals of the battery to transfer this energy. Is this calculation invalid? If not, how do you interpret this calculation in terms of electromagnetic waves?


With respect to your battery, when it provides one Coulomb of charge, oxidation in the anode really does cause 6E18 electrons to leave negative terminal, and a further 6E18 electrons to enter the positive terminal where they reduce a similar number of ions. 

The point being made in your source is that it isn't the same electrons leaving one side as arriving at the other.  Electrons are all interchangeable, and the ones going into the positive terminal probably started very, very close to where they ended up.  You can think of electric field as pushing one electron a slight distance, which repulses another electron, which repulses another... until finally far down the chain one electron will reach the positive terminal and reduce an ion. 

In addition, you can apply the same reasoning to the movement of ions in the battery cell itself, which also move under the effect of the cell voltage:



Taken from wikipedia's article on galvanic cells.