This is indeed not a fundamental problem, because classical electrodynamics is only an approximate description of electrodynamics valid in some appropriate scaling limit. If one then attempts to formulate such an approximate theory completely divorced from the real fundamental laws of Nature, then one typically encounters problems when naively taking limits to scales where the theory should not be valid.
A non-trivial problem in classical electrodynamics where renormalization is needed is the rigorous treatment of electromagnetic radiation. While one can calculate the emitted radiation of accelerated charges without problem, the backreaction of the emitted radiation on the charges is typically ignored. E.g. the OP's book contains a calculation of the emitted radiation by pulsars, it mentions that these pulsars slow down due to conservation of energy, but it's not pointed out that this implies that the Lorentz force equation as mentioned in the book must therefore be wrong as it doesn't contain any terms that could exert a torque on a magnet that is freely rotating in empty space.
What needs to be included, therefore, is the force exerted on a charge due to its own electromagnetic field, but this is divergent for point charges. A rigorous treatment of the self-force was only given recently in this article.
In general, analogues of the renormalization group methods as used in QFT will have to be invoked in all physics disciplines where the degrees of freedom reside in arbitrarily small length scales (e.g. fluid dynamics), as soon as one is rigorous enough. The fact that we typically don't see such methods applied in textbooks is because they typically don't treat the subject in a rigorous way.