That's correct.
FFT has is right. The block does not move all at once. The far end of the block is at rest until the information that you pushed it can reach it, which in a solid or 'rigid' body is always slower than the speed of light in vacuum.
Yes, the rod has been very slightly deformed, and the force of the push propagates down its length as a sound wave (a distortion of the structure.) So the concept of 'rigidity' is always slightly wrong -- there is no such thing as a perfectly rigid body in nature.But I think there is an interesting issue here appart from this. Would you say that the rod has been deformed? or is acting as a deformed body? or is totally rigid but in a very special geometry (space-time)?. With this I mean for example that if you consider the fact that simultaneity can be broken could you consider from the frame of reference of the traveling perturbation that the rod is moving as a rigid body? Wouldn't from that frame of reference all the parts of the rod moved at the same time?
You could consider the frame of reference of the wave propagating down the rod. If the wave propagates at the speed of light, then it is correct that the whole rod moves all at once according to you. But then the rod's length is also infinitely short, because of the relativistic length contraction! You will also find that clocks fixed to the ends of the rod read very different times -- a synchronized network of clocks in the rod's frame are not in sync according to you!






