Discovering that the Maxwell equations can be written in this succinct form is kind of mind-blowing, but I'm wondering whether it provides any additional insight? It seems like one needs to do a significant "unpacking" to actually use the equation or gain insight from it. I would love to hear an explanation of electrodynamics starting with "star ddA = J".
Yes, I kinda agree. The underlying physics is, by definition, the same, and what you gain by reducing the number of equations you lose by having more complicated "objects" and needing more advanced maths to handle them (e.g. the electromagnetic field tensor, external calculus and whatnot vs. just vector fields and basic vector calculus).
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.
You get new insights, that F is a curvature 2-form. Written in this way it's explicit that EM is also a geometric theory. This observation opens the door to Yang-Mills theories which are behind all the Standard Model.
