Dyson's statement was about Maxwell's theory, not all of E&M and its interactions with every other physical theory. There are no point charges in Maxwell's theory, for example.
I think the point is that electromagnetism really is a very complex and knotty phenomenon, and we moderns are spoiled by having
(1) Heavyside's deceptively elegant formulas for the theory (which we now call the Maxwell Equations) and,
(2) A well oiled pedagogy about how those equations the capture specific phenomena named after the likes of Ampere and Faraday, and
(3) Powerful formalisms for studying circuits, optics, transmission lines etc that can mostly be used without going back to the fundamentals.
Most of that was not available to Maxwell when he started out, so it's not surprising that he presented his theory in a way that reflected the real complexity of it all.
(1) Heavyside's deceptively elegant formulas for the theory (which we now call the Maxwell Equations) and,
(2) A well oiled pedagogy about how those equations the capture specific phenomena named after the likes of Ampere and Faraday, and
(3) Powerful formalisms for studying circuits, optics, transmission lines etc that can mostly be used without going back to the fundamentals.
Most of that was not available to Maxwell when he started out, so it's not surprising that he presented his theory in a way that reflected the real complexity of it all.