That seems like it probably is the right way to do it.
Since I was going by dim memories of Jackson I hadn't even heard of the "reduction of order" approach, which sounds actually like a reasonable thing to do, and in this paper they seem to be giving it a justification on more fundamental grounds. The runaway solutions and pre-acceleration and such always seemed like they had to be artifacts, given that they were starting with a theory that was supposed to be causal and have energy conservation.