This is all a discussion of a more general situation, but distorted by assumptions based on personal observations of bulk "reality" that don't extend to the scale in the original question.
Energy can be input into any material in the form of vibrations. They may be some continuous vibration source (whether containing single or multiple frequencies), or they may be created by a single impact which causes a broad range of frequencies but is not sustained over time. For OP's thought experiment, the material in question is in the form of an extremely long rod.
Every real material we know has elastic properties best modeled in mathematics as a complex number which changes depending on frequency of the mechanical shock wave or vibration traveling through it. The "real" part of that complex number describes the velocity of the wave, which is NEVER infinitely fast. In fact, not surprisingly, the propagation velocity is MUCH slower than light, and even slower than the velocity in air. In OP's experiment, this velocity is what determines how long it will take for the wave inserted at the earth's end of the rod to reach to other end. It will be a VERY long time, MUCH longer than the time for an electromagnetic wave (light) to travel the same distance through a vacuum. The other part of this complex number, the "imaginary" part, describes how the wave amplitude is attenuated as it travels though the medium. That is, some of the energy of the wave is absorbed by the medium itself and NOT transmitted along the rod. Even with a very tiny fraction of attenuation present, the extreme length of that rod means the amplitude that makes it all the way to the other end (and yes, it WILL be greater than zero) will be so small that detecting it will be extremely difficult.
Why should this be? The material we are using may be a pure metal or metallic alloy composed of just metal atoms and their electrons in shells, with some of the outermost electrons so loosely bound to their original hosts that they have enough of their own energy (at ANY temperature greater than absolute zero) to move among the atoms as "free" conduction-band electrons. Or, the material may be composed of other non-metallic molecules in which the electrons are more tightly bound and there are no free electrons in conduction bands. Either way, the molecules or atoms each are at one place in space on average, but they all vibrate around that point slightly, and so do all their neighbors. The vibrational motions of each influences its neighbors, and in this way they all kind of share their energy so that the whole mass of material has an average energy content per atom (or molecule), and some variation (distribution) of energies.
Now, what happens when a mechanical wave or impulse is applied to the molecules (atoms) at one end of the long rod? It increases the natural vibration motion of the atoms at the end in one particular direction, and they move in response. This disturbs them from their former equilibrium point, influencing the immediate neighbors and pushing them from their original positions. A "chain reaction" of movement propagates the wave or motion along the rod. Even if the actions are completely elastic - that is, ALL the energy in the initial impulse is passed on down the rod - the process takes a finite time because the mechanism depends on the speed at which the atoms (or molecules) are naturally moving and colliding (well, not really - more like influencing) with each other. But what happens in reality is that some of these interactions leave a little bit of that initial energy with the current atom - it does NOT all get passed on the the next one. So the current atom is left with slightly more energy than it started out with. At some later point in time it probably will pass this on to another atom nearby in the normal natural "sharing" process, BUT that new receiver will NOT be the one in the direction of the far end of the rod. It will be in some random direction. At a macroscopic level, we observe this increased energy content in the rod as a temperature increase. That is why simply pounding on a piece of metal can heat it. This is the mechanism that produces the reduction in the amplitude of the initial mechanical wave as it travels along the rod - the phenomenon we label attenuation of the wave. It is unavoidable in the real world.
In our own personal experiences of "reality" we do not notice that it takes real time for a whack on the end of a rod even 100 feet long to be felt at the other end. And of course we don't appreciate that the jump of the far end of the rod is just a little less than the jump at the whacked end. So we don't recognize these effects in our reality.
The exact details of the velocity of propagation of a mechanical wave in a solid material, and the magnitude of the attenuation effect, depends on the details of the material from which the rod is formed. That comes down to the natural vibrational energy states of the atoms or molecules in the rod and their own natural vibration frequencies. The result is that the impact of these molecular-level factors varies according to the mis-match between their natural frequencies and the frequency of the mechanical wave being transmitted through the rod. So the process ends up impacting the transmission of that mechanical wave in three ways: it determines the average propagation speed, it determines the attenuation, or reduction in wave amplitude along the rod, and it makes these changes to a different extent for each frequency in the original wave mix. The result at the other end of the rod is that there is a finite "signal" transmission time, it is weak, and its details will be substantially altered by distortion of the original multi-frequency waveform.
Sorry, but trying to escape from the characteristics of wave propagation through space (light) actiually still leaves us with wave propagation of a different kind through a different medium, and the limits on it are more than the limits of light in a vacuum.