I don’t think that the paradoxies of time travel that arise with the closed timelike curves of superluminal travel are a sufficient reason to rule out the possibility of faster than light travel. Even if FTL enables time travel, time travel might still work. To avoid all the paradoxies of time travel, it’s useful to assume that the many world theory is true and you don’t jump into the past of your own universe, but the past of a very similar universe. Whether you can actually go back to your original universe is a different question. Maybe that’s not possible.
And the conclusion of that would be that FTL travel would enable travelling between different universes. That would be pretty cool, actually.
All the scientific proposals for FTL drives involve a hypothetical substance called exotic matter. As far as I know there is no evidence of something like that existing, unless it’s actually the stuff that dark energy is made out of. But even if somebody built an Alcubierre drive with exotic matter, I wouldn’t be sure that it would actually enable FTL travel. After all, it requires the manipulation of spacetime. But information can only transmitted with the speed of light within spacetime. The detection of gravitational waves moving at the speed of light indicates that strongly. While it’s possible to warp spacetime, the warp only moves with the speed of light.
To avoid that problem it may be necessary to place space time warp devices along a defined track and trigger them in a way that makes it appear like a signal was moving faster than light. It might be possible to build something like FTL trains that move cargo between different star systems, but the energy and matter requirements for such FTL tracks might make them practically unfeasible. It might be possible to build such tracks within a star system, though.
I think it’s unlikely that FTL travel works within this universe. Dark energy is more likely an inherent property of spacetime and not a kind of substance that can be manipulated with technology.