I asked the smartest engineer I ever knew how many threads do I need to engage for full
strength and he just said "3". So I'm with you there. But that's for full strength of a mechanical fastening against sheer/pull. How does that relate to chucking a rod I don't see. A straight rod does not become a spring when held by a clamp parallel to its surface. or I don't see how.
Where is that coming from? The more clamp in contact with the rod the better the grab which is what chucking is all about -holding the work firmer than the tool trying to stop it.
The best high-quality springs are rods (quartz fibers) in torsion. To make them deflect more, you just use a longer
length. Angular deflection threatens cut quality, and keeping the workpiece nearly rigid means holding
it near the cut. The screw-thread issue is about stretching (also elastic deformation, but different geometry).
Simple (good-but-not-perfect) physics model of friction is that the static (holding force) friction is a coefficient
times the force normal to the surfaces in contact. Area doesn't come into that model; as long as the forces
don't significantly deform the surfaces, it's theoretically NOT better holding/more friction to use larger area, only
to use higher normal (clamping) force. This implies one should use a clamp caul with enough contact area to
avoid plastic deformation, but NOT to otherwise maximize area.
Much of the 'normal force' in threaded objects is from the geometric ~30 degree slope of the contact surface; that
roughly doubles the normal force and friction, compared to cylinder-in-cylinder clamping.