The origin of Maxwell's equations is completely phenomenological at a macroscopic level. In terms of fields only, as it was shown by Einstein in 1905, they encode special relativity as the underlying symmetry and, moreover, they are linear partial differential equations of first order in the fields. Moving to the Lorentz-covariant Minkowski spacetime approach, whose Lagrangian implementation necessarily brings into picture the 4-potentials as an equivalent description of the theory, the only admissible (i.e. second order in the derivatives of the potentials) kinetic term in the Lagrangian density is (up to a conveniently chosen numerical factor) $F_{\mu\nu}F^{\mu\nu}$. Encoding field-sources (stationary electric charges and moving electric charges/currents) in a tensorial object such as $j_\mu$ and assuming minimal coupling (here one can prove that a minimal coupling conserved current-potential is a must), one finds that:
$$ \partial^{\mu} F_{\mu\nu} = \kappa j_{\nu}$$
is the equivalent of Einstein's equations of GR, along with $\partial^{[\mu}F^{\nu\sigma]} = 0$ which is required by the antisymmetry of the Faraday tensor, in turn a consequence of having the 4-current conserved.
One further point. We can try to deform $ \partial^{\mu} F_{\mu\nu} = \kappa j_{\nu}$ to, let us say $ \partial^{\mu} F_{\mu\nu} + a \partial_\nu \partial^{\mu} A_{\mu} = \kappa j_{\nu}$, but at the price of losing both current conservation, and the link to the phenomenological equations in terms of $\vec{E}, \vec{B}$.