Not quite.
For one thing, typically the Lieb-Robinson bound is not tight, that is there is a slower velocity (of sound, for example) which controls the spread of information. The Lieb-Robinson bound is really a microscopic construction that knows very little about the actual dynamics of the system.
Furthermore, even in systems where the Lieb-Robinson bound is obeyed, the system does not have to obey special relativity. There is still a preferred frame set by the lattice and sometimes this can play a crucial role in the physics. Of course, to be fair sometimes Lorentz invariance does emerge, but this is a complex dynamical phenomenon not captured by Lieb-Robinson.
Finally, the Lieb-Robinson bound permits violation of the "light cone" by exponential tails. To the extent that one wants to believe the relativistic light cone is sharper than that, one needs more than Lieb-Robinson. Since this is the Beyond the Standard Model section, it should be mentioned that it is not clear exactly how seriously to take the relativistic light cone especially if, for example, geometry itself is fluctuating.
Hope this helps.