I go by Mark Wood's rule -- thread that halfway into the hole. Two rod diameters is probably a good figure of thumb, too.
Usually with fasteners you're just interested in clamping two parts together. That's not really the case here -- you're asking the threaded connection to locate the ball link along the length of the rod and keep it pointed straight along the rod. It's that "keeping it pointed straight" that, I think, would be the first thing to cause problems. If the link can wiggle and rock on the threaded rod, then it'll wear, and then every bad thing that can happen with wear might happen.
So I'd ask if the fastener is going to have a chance to wiggle in the hole, or in the case of those plastic shanks, will the shank bend. If the answer is "yes", then you need more thread engagement.