I'm not so sure. Usually you take the four-potential either in Lorenz or Coulomb gauge. In the Lorenz gauge, using Cartesian coordinates, your four components are decoupled and obey the wave equation. Then you choose the retarded solution for these waves. In Coulomb gauge, using Cartesian coordinates, you get a Poisson equation for ##\Phi=A^0## and decoupled wave equations for ##\vec{A}## but with a modified current density. Again using the retarded solution for the latter, you get in both cases the same result for the electromagnetic field (i.e., Jefimenko's equations), as it must be, because Lorenz and Coulomb gauge potentials are just mapped to each other by a gauge transformation.
On the other hand, you can choose any other not so common gauge. I can't find it at the moment, but there was once a very illuminating article in Am. J. Phys. where the author showed that you can define gauges, where part of the potential propagator at either speed you like (in Lorenz gauge all components are retarded with ##c## as the "signal velocity", in Coulomb gauge the scalar potential (temporal part of the four-potential) is intantaneous), all leading to the correct Jefimenko solutions for the field components.
Of course, there's no principle way to argue for the one or the other gauge (except simplifications to find proper solutions for a given problem) how the four-potential propagate, because they always contain unphysical degrees of freedom, which precisely cancel when calculating the electromagnetic field from the potentials. For the electromagnetic field components however, you can argue that it must be retarded with ##c## being the phase velocity of all field components, because these are observable fields. So I think, it's indeed pretty straight forward to derive Jefimenko's equations directly from Maxwell's equations. I'll try to do this later today.