I think your reading too much into this. We can manipulate objects at a distance with magnetism. We can manipulate charged particls in an electric field. We can predict orbits in a gravitational field. Now scientists are manipulating particle states at a distance. All this stuff is "action at a distance." For all the clever calculations and speculations scientists make, they still can't tell us the underlying mechanics of how forces act at a distance. It's spooky, erie, otherworldly for the time being.
It is, but it's not all quite the same kind of "action at a distance." The correlation of wave functions in quantum mechanics happens without intermediaries, and at superluminal velocities. It is "instantaneous" in fact, although that phrase actually has no Lorentz invariant meaning.
For all the clever calculations and speculations scientists make, they still can't tell us the underlying mechanics of how forces act at a distance.
This statement is false.
We understand quite well how forces act at a distance, and quantum teleportation does not involve those mechanisms. Fundamental interactions are mediated through particles called gauge bosons. The gauge particle of the electromagnetic field is the photon. In order for an electric field to exert a force, the charged particles involved exchange photons. There are gauge particles for each of the fundamental interactions: gluons (strong nuclear force) W and Z bosons (weak nuclear force) and gravitons, in addition to photons. There is some controversy about whether gravitons are actually gauge particles; we do not have a quantum theory of gravity.
All of those gauge bosons provide attractions or repulsions at the speed of light.
The correlation of coherent quantum particles has physical consequences, but is not one of those interactions. It is not a "force" and does not occur at light-speed. This is the essence of Einstein's objection to it.