In 1943 CJ Eliezer published a paper claiming that the self-force prevents a zero angular momentum particle from ever reaching the center of an attractive Coulomb potential (and what's more that it can collide with a repulsive potential). As stated in the paper this result is somewhat counterintuitive, but the reasoning seems like a relatively straightforward differential equation argument.
In thinking about this the understanding of the self-force that I came to is that while you can derive it purely from energy and momentum conservation (and thus it must be valid in any theory of classical charged point particles), the resulting differential equation is better thought of as a consistency condition than equations of motion (i.e. only solutions of the 3rd order self-force equation correspond to point particle sources that have solutions to Maxwell's equations that lose or gain the correct amounts of energy and momentum at the point particle corresponding to its motion). And while one would like to be able to define a dynamical system of a point particle coupled to the electromagnetic field with physically plausible boundary conditions, even eliminating runaway solutions doesn't prevent you from being forced to include waves coming in from past infinity (i.e. in pre-acceleration solutions, which do not have to have runaway behavior, the particle will move before an external force is applied, meaning that it must be gaining energy from radiation from past infinity).
Assuming that Eliezer's result is correct it seems like every trajectory of a particle in a stationary Coulomb potential similary requires radiation adding energy to the particle (otherwise you can prove with a simple energy argument that it must fall in). So the question is, is my interpretation of the dynamics of the self-force correct and is there a physical or intuitive explanation for this extremely pathological behavior in the presence of a Coulomb potential?