Earlier quoted context omitted.
Like QM, doing it analytically clears away misconceptions and gives a consistent view - but it isn't exactly something you can teach to children. It's useful to be able to explain things to laypeople and the not mathematically inclined, which means tightening up the metaphors.
Also Maxwell's equations only give you part of it - for the question of why an atom's electrons don't just fall into the nucleus you need QM. And then you get into the self energy of the electron which is not explained by any present physics ( http://www.feynmanlectures.caltech.edu/II_28.html ). So you can't just do the math because we don't have all the math.
You won't really run into the atom stuff until you try to do something like calculate dielectric constants analytically, though self-energy is definitely a pretty easy trap to fall in even for macroscopic stuff.