It's simply wrong to extrapolate like this. In the early universe, matter was very uniform and Hubble's law held very accurately. Today, matter at small scales has clumped, and the clumps have been interacting gravitationally in complicated ways for billions of years. Their velocities are functions of that complicated interaction history. Only when you average over very large scales, large enough that matter hasn't had time to clump significantly at that scale, do you still see signs of the original uniform motion.
Two lumps of matter one meter apart don't "want" to be moving relative to each other at the Hubble speed or any other speed. They move according to the forces of gravity, electromagnetism, and so on, just like everything else. They are "everything else". There's no cosmological force on top of the familiar forces that is trying to make things move at the Hubble speed.
It seems to be a very common misconception, even among professional physicists, that there is such a force. I'm not sure why, but my best guess is that a lot of the time when people do calculations in general relativity, they're just calculating geodesics of particles in fixed spacetime backgrounds, and after a while they seem to forget that GR isn't just a theory of geodesics in arbitrary spacetime backgrounds, but is actually a theory of gravity. Then they assume the FLRW geometry (the approximate shape of the universe at large scales) is valid at small scales, because they think of it as just a background, and they calculate local geodesics in it and find a force. They don't notice that by assuming the FLRW geometry, they also assumed (via the GR field equations) that space is filled uniformly with matter like it was in the early universe, and the force they calculated is the gravitational influence of that matter. In reality, that matter isn't there (it's elsewhere, having clumped into stars and planets and whatnot), and so the force doesn't exist. The only real gravitational forces are the forces from the matter that's actually present, in its actual location. The "Hubble expansion force" is a made-up force arising from matter that doesn't exist via a misunderstanding of general relativity.
