I just learned from the existence of the theoretical Lieb-Robinson bound, which indicates that the speed at which information can be propagated in non-relativistic quantum systems cannot exceed this upper limit. If this is true, then does it mean that the entanglement is not transmitting information? So what does it mean for protocols like BB84 (quantum key distribution)?
Entanglement does not transmit information, as follows from the No-communication theorem.
Lieb-Robinson bound is a limit on speed at which perturbation propagates using short-range interactions, for example in spin lattice. I doubt it means something for protocols such as BB84; you can transmit quantum information by sending polarized photons, and photons move at the speed of light; or you can transmit quantum information using quantum teleportation protocol, also limited by the speed of light.