The point is actually in even transferring an arbitrary quantum state from one quantum particle to another, and given an entangled pair existing in advance. Transferring a quantum state from one particle to another isn't an easy or obvious task, because you can't measure it or you would collapse it.

Note that "classical" is just stating that a classical channel is good enough to serve that purpose. A channel that preserves quantum state can of course be used, it's just that that isn't required for this to work.

All classical phenomena are quantum, we just use the word "classical" to describe subsets of quantum phenomena acting qualitatively in ways that are consistent with macroscopic phenomena.