If I recall, I think the shaft is actually pentagonal, not hexagonal. Drilling and tapping a set screw may work, but you would have to go through the entire body of the "upper" plastic mechanism to do so, and downstream, accessibility would be an issue as well. With that solution, one would also have to remove the ceramic grinder at the other end entirely to fill up the grinder with pepper corns, salt, etc.
I did a wine bottle grinder with a Crushgrind and the drive end (where the shaft is inserted into the drive mechanism in question in the original post) is embedded/glued into the top of the wine bottle. Once done, that drive mechanism was no longer accessible. Only way to fill it was to remove the ceramic grinder at the other end. In retrospect, I needed a spigot on the wine bottle portion that the top of the bottle could ride on (to align the top with the body). With that the top could be removed to fill. Lesson learned!