"we cannot say that we understand all there is to know about electromagnetic phenomena because we can make a few simple predictions" This is quite an understat

"we cannot say that we understand all there is to know about electromagnetic phenomena because we can make a few simple predictions" This is quite an understatement. QED, which is the underlying theory of EM, is the most accurate scientific theory ever. Full stop. It's theoretical predictions match experimentally measured observables out to ten decimal places. By your standards we don't really understand any physical phenomena. To bring up your analogy with the sun: our understanding of QED is like saying that we know the exact chemical composition of the sun to an accuracy of one part in ten billion. I think we would be justified in stating that we understand the sun in those circumstances. On a related note, QED does provide a lot of context for what "light"(photons) is(are). It's the gauge boson (read: mediating particle) of the electromagnetic force. In my mind it doesn't get much more elegant than that.