Physical theories are supposed to work together, and classical mech and electrodynamics mostly do. Electrodynamics is expressed in the language of Newtonian mechanics (e.g. fields represent force per charge), and the experiments that established the laws of electromagnetism used Newtonian mechanics as their framework. There are specific inconsistencies between Newtonian mechanics and electrodynamics, inconsistencies which spurred the development of special relativity.

Physics theories are not supposed to be mutually incommensurable, like the ideas of two philosophers from different schools who don't understand each other. Physics doesn't work that way, and physicists have never been satisfied with that.