Causality violations probably aren't really that much of a problem for mathematical physics. Geroch (a very well known and highly cited relativist) takes a shot at this in a short paper considering a "democracy of causal cones" rather than simply the one that matches light: "All systems, and their cones, are on an equal footing: No one set of fields, or one set of causal cones, has priority over any others." Geroch's argument is that one can still do initial-value-formulation physics even in the presence of causal cones whose width is different -- including wider -- than that of light.

https://arxiv.org/abs/1005.1614 -- it is highly technical, but has this as the kicker:

"we would then insist that the final field configuration is, not what the observer wills it to be, but rather what follows, evolving these data via [dynamical equations]"

Notably, this is backwards from the Alcubierre drive's formulation, which starts with a desired final field configuration, and is trapped by having no "before the bubble existed" initial values surface. The Alcubierre solution is eternal, and offers no way to be stitched into our apparently non-eternal universe (it sure seems to have a beginning, or at least an early hot/dense boundary close to which the presence of an Alcubierre bubble seems even less physically plausible than today).

Although it is safe to describe really fantastical configurations of matter very near that early boundary, one has to settle things down in far less than seconds, in order to match several different lines of evidence of about the distribution of the first free-streaming photons (and the earlier free-streaming neutrinos, when we figure out how to build observatories to study them in detail). You could put Alcubierre bubbles everywhere during Cosmic Inflation, for instance, but you have to very quickly and aggressively suppress their observables.

Let's turn this on its head: we can in principle construct a spacelike values-surface at time today and let it evolve. A problem we have with this is that the presence of either or both of not-yet-formed or formed-but-eventually-evaporating black holes makes a complete calculation of some far future values-surface impossible in principle, as far as we know, which means in the absence some new fundamental theory of quantum gravity that has yet to be discovered. Putting an Alcubierre drive onto the same today values-surface likewise blocks the complete calculation of the future values-surface, in principle. Dusting the cosmos with lots of Alcubierre drives makes things much worse. But maybe that's OK, because the cosmos appears to be dusted with lots of black holes and configurations of matter (like our own Virgo cluster) that will inevitably mostly settle into giant black holes that may one day evaporate. This means we can't really make a concrete prediction about the far future of our cosmos, in principle. I wouldn't sweat it, because we can't set down much more than the barest approximation of the real values-surface in practice anyway.

So since we already have problems with causality thanks to gravitational singularities (or if you prefer, apparent trapping surfaces), and we have excellent evidence of them existing in our universe, an argument that "X poses problems for causality and therefore must be unphysical" is not especially compelling.

Instead, the problem grinds down into [a] laying out an initial values surface gets harder, especially if we can't rely on a programme of pushing it further and further back in time so that unusual configurations of matter (black holes, Alcubierre bubbles) have the ability to develop through dynamical laws (which is what we do with physical cosmology, for instance); and [b] writing down the dynamical laws that relate neighbouring values-surfaces (which gets harder because we need more than the Lorentz symmetry group and GL(4).

It might be fun to try to make [a] and [b] work in the presence of various FTL mechanisms, and indeed to some extent we do this in very specific settings (cosmic inflation, for instance). However, from the time of the CMB to now there is nothing to suggest a need to account for FTL material (or normal material embedded in a dynamical spacetime that moves it around FTL-style) at any scale, from the cosmological to the laboratory. The theory would undoubtedly be much more complicated than what we already have (which has already evolved and become more complicated as more evidence has become available), so there would not be much point to it.

Finally, two extracts from the Geroch paper:

"[I]n a similar vein, there exist solutions of Einstein’s equation in general relativity that manifest closed causal curves. But we do not ... allow observers to build time-machines at their pleasure. Instead, we permit observers to construct initial conditions — and then we require that they live with the consequences of those conditions."

which is exactly the opposite of Alcubierre's approach in https://arxiv.org/abs/gr-qc/0009013v1 where he starts with a time machine first, and never describes the initial conditions before it exists. One would usually expect that for any artifact such as a spacecraft, there is a time before which it is fully built and switched on. One might compare that with an eternal Schwarzschild black hole and a black hole that forms via gravitational collapse of a massive star (or direct collapse of matter in the early universe).

And, "from the present viewpoint the problems associated with superluminal signals do not seem nearly as severe as they did at first glance", a statement that is likely safe should such signals ever be proven.