The total tension at the two ends of your rod (and at any location between the two ends) must me constant (or otherwise, the rod would move, as there would be a net force). Therefore, tension per unit area is smaller at the wide end than at the narrow end. Since elongation at each location depends not on the total tension, but on stress (tension per area), it follows that if you pull with a given force a non-uniform rod, the smaller cross-section portions will elongate more than the larger cross-section ones.
Now, in your question, you compare the tapered bar with a bar that has a radius that is the average radius, and you get a deformation that differs from the deformation non-uniform bar. This is because stress depends on area, therefore radius squared, not radius. Narrow portions elongate a lot as the r squared scaling concentrates stress much more than the linear r scaling of the uniform bar with "equivalent" average radius in your question.
You can prove this mathematically, but to see why intuitively, imagine an ideal cone that can be deformed elastically as much as you want without being destroyed plastically deformed. At the tip, the stress is infinite. That is, at the very tip, with zero area, deformation is infinite, and the cone would elongate by an infinite amount. Of course, this is not physically feasible, but limit cases often help in understanding the physics of a system. If R is the radius of the wide end of the cone, replacing the cylinder by a rod of uniform radius R/2 does not allow infinite extension.