You can even write them in terms of the four-vector A rather than F (related by F = dA) to reduce them to ddA = J.
To get an intuition of the physics, I think the traditional 4 equation form is actually more useful, as you can construct toy examples and study the equations one at a time in isolation.
Where the more advanced formulations are useful, and actually are used, is for stuff like relativistic physics where 4-vectors, curved spacetime etc. are needed and not just a gimmick.
But for more down-to-earth applications of electrodynamics like antennas, field propagation in various forms of matter etc., the classical version is fine.
*d*