Maxwell's equations from continuum limit

I think you're interpreting him a little too literally. His meaning can be taken more literally if you use his first example, a wave equation for sound waves in a solid. I don't think he literally means that you can get Maxwell's equations by coupling together massive particles with springs -- when you do that, you get sound waves, not light waves. There are fundamental reasons why you can't hope to get a description of the electromagnetic field by literally using this method:

(1) The speed at which wave disturbances propagate will be less than $c$. (This holds if the matter that the springs and masses are made of are any reasonable form of matter -- specifically, one that obeys the dominant energy condition.)

(2) You're going to have a hard time getting rid of the longitudinal vibrations and getting the transverse ones. The prohibition on longitudinal vibrations really only makes sense relativistically, for a wave propagating at $c$.

(3) You will essentially be constructing an aether theory, which will not be consistent with Maxwell's equations (or at least, not with how we interpret them nowadays).

The discrete degrees of freedom that he has in mind for the EM field are probably more like the normal modes of a cavity, or something of a similar nature in QED.