Arguments like A) or B) are formalistic exercises that start with "simple assumptions" about functional and transformation properties of field action and pretend that those inevitably lead to the correct equations of motion for the field. In reality, those arguments assume additional things that are rarely mentioned, for example that there is only one universal field $A^\mu$, or that the field is not too singular function of position (no point particles), or that the Lagrangian density is Lorentz invariant (not necessary). These arguments have no predictive value in case of EM theory, they are created in reverse, with knowledge of what the correct result is.
The actual first principles based on generalization of experience are that 1) EM field obeys Maxwell's equations and 2) particles experience the field according to the equations of motion including the Lorentz force formula. Then the simplest form of action that results in Maxwell's equations and particle equations of motion is sought.
The action terms look like what you wrote above because that is one of the forms that results in Maxwell's equations for fields for prescribed behaviour of charge and current distribution. It does not give equations for the charge and current distributions, those are assumed to be known.
I wrote one of the forms, because there are different ways to formulate the Hamilton principle for the EM field obeying Maxwell's equations. Just as in classical mechanics, many different Lagrangians can give the same result.
The formulations can also differ in what the independent variables are, and what the action function is. Some are gauge-invariant like $F^{\mu\nu}F_{\mu\nu}$, some are gauge-variant such as $\partial_\mu \! A\, \partial_\nu\! A~~\partial^\mu\! A \partial^\nu\! A$$\partial_\mu \! A_\nu~~\partial^\mu\! A^\nu$.