You have succombed to the "perception plague" of science; your intuition is biased by everyday "normal" perception.
--Time seems to flow absolute throughout the universe, two simultaneous events happen at the same time for everybody, light seems to travel with unlimited speed, particles and waves seem like perfectly distinct entities (you're either one or the other but certainly not both), no one object can be in TWO DIFFERENT CONTRADICTORY STATES AT ONCE! (schrodingers cat dead/alive)--
... these are just a few examples of the things physics knowledge has had to PRY away from our stubborn brain, who has always learned the ways of nature through the illusions and trickery of sensory perception.
My point is... that if you push a rod on Earth, the other end seems to reply at an instant. But this is only an illusion created by using a small rod, just like "instantaneous light" by flipping a switch, is only the illusion of being at a short distance from the bulb.
Instantaneous response is simply not possible, a delay can no longer be pushed aside as "negligable" if the rod were of a length where the speed of light is insufficient to traverse the rod in a noticable duration of time. If a rod were that long, a noticable difference MUST OCCUR between initial push and final lunge forward. What happens is that there are gaps between the atoms in the rod, and when you push one end, these gaps become compressed at that end. As the compressed end tries to establish euqlibrium once again, it pushes back in all directions, even against your finger; you feel this as inertia. Since your finger pushes back in one direction, the compression is forced to expand into the opposite direction, which causes other atoms to be compressed which in turn expand and push other atoms. It's like a longitudinal wave is traveling along the rod whilst carrying the signal that "one end has been pushed", and therefore in an effort to re-establish equalibrium, the other end must also move forward. Since this signal is comunicated as a compression/rarefaction wave front, it stands that it is subject to the same rules of relativity as is the rest of wave mechanics.
If we could watch this on a tape in slow-mo, we would see a wave resembling that of sound, plow through the rod and end up at the tip as a "leap forward". This happens in normal sized rods as well. But the speed of propogation is so quick, and over a short distance, that our brain doesn't take notice. Since rigidity determines the elasticity of the medium, and the medium in turn determines the "response time" for pushed atoms to jerk back (speed of propogation), then it follows that a PERFECTLY rigid rod would be required as you said above... which is simply science fiction. There is no such substance in the MATERIAL UNIVERSE from which a rod of INFINITE rigidity can be forged. Therefore this thought experiment is not consistent with established theory, and can never be promoted to the status of "technology."