Over time, won't the chances of grazing reduce-- they eventually slip into orbits too distant from each other, fall to the host star, or swing out until they escape the local gravity well?
yes, as I understand it, the process described above is indeed unstable, that means, if it occurs in nature, then it will either eventually escape the host star or fall into the host star.
However, I believe that with additional intelligent regulatory circuitry, the process could be regulated well enough such that the process remains within certain boundaries, i.e. practically stable.
Such a process could be used to supply humans with energy for an indefinite time, long after the sun has burned out its hydrogen. It does, however, require intelligence and some bit of control. I have left out the description of regulatory circuitry in the above description for brevity.