The point is that causality violations don't happen just because of FTL, and that a good choice of values-surfaces reveals that.

"Preferred" foliations are perfectly fine in relativity when picked out by the matter. Working relativists do it all the time. One just has to be careful about preserving Lorentz-covariance for observers whose time axis is badly misaligned from the "preferred" one, because the (invariant sector of) physics must be identical for them. In particular a good choice of time axis along which to split spacetime values into time-ordered spatial values is a thermodynamic arrow of time that applies to the vast majority of the matter subject to dynamical equations.

Example preferred foliations are the cosmological scale factor, and terrestrial atomic time like UTC; they are picked out by the symmetries of the matter (large scale homogeneity and isotropy, near-sphericity and near-rotationlessness, respectively). "Lab frames" for example, are often UTC. That relativity provides mechanism to transform tensor-components as one switches systems of coordinates (or frames) is a great freedom.

The point is that in e.g. the cosmological foliation, an FTL (but not instantaneous) traveller will only be at one location on a cosmlogical values surface.

Geroch's point is that one should just accept this picture and calculate the evolution of field-values in the presence of objects with causal cones of different widths. There is no causality violation with very gentle assumptions about the spacetime. There are likely to be weird observations though, peculiar to the observers. But we can already have weird stroboscopic effects and so forth for merely very fast (not even relativistic!) travellers thanks to thinks like the persistence of vision or analogues in cameras and detectors.

However, an instantaneous traveller (>>>>> FTL, literally infinite speed and causal cones that are infinite in width) may mathematically appear up to infinite times on a single spacelike hypersurface. This does make a mess unless one can extract from the FTL traveller's configuration an equation of motion with enough constraints to make predictions. (For instance, if you have too many copies in the same "slice" of spacetime, does the entire spacetime collapse into a black hole? Can the copies be inside each other? Or only the bosons? And so on.)

Finally, a backwards time traveller need not even do FTL. Again, it will appear twice (or more) on well-chosen values surfaces, posing no problems for causality analysis. Here we return to the question of the thermodynamic arrow-of-time.

Let's consider a Rocket(with some amount of fuel) accelerating (gently) to the right, leaving an exhaust, that an intertial observer ticking at t watches :

t_0 R(10)
t_1 eR(9)
t_2 eeR(8)
t_3 eeeR(7)
t_4 eeeeR(6)

in this small picture one would pick out a thermodyanmic arrow of time pointing downwards, because the rocket+exhaust's Boltzmann entropy is increasing. The fuel is well-ordered in the rocket's tank, and much more disordered in the exhaust.

But the dynamical equations in this arrangement of matter is perfectly reversible. Let's complicate this a bit:

t_0 R(10) eeeeeeeeeR(1)
t_1 eR(9) eeeeeeeeR(2)
t_2 eeR(8) eeeeeeeR(3)
t_3 eeeR(7) eeeeeeR(4)
t_4 eeeeR(6) eeeeeR(5)

Here we have what time travel looks like with this foliation: A rocket behaving normally, and a rocket that is picking up fuel from a hot exhaust, storing it orderly into its tank.

R, moving gently through space, sees the thermodynamically-opposite copy out the window in both directions. R believes there is a weird causality violation, because one copy of R sees a copy of itself with less fuel (and wristwatches showing times in the future, but ticking backwards).

However, t just sees a time traveller, where the backwards time traveller has an opposite thermodynamic arrow of time.

Now we make t the entire cosmic microwave background in its cosmological rest frame, cooling towards the down direction. R can measure the temperature (and dipole redshift from the gentle acceleration) of the CMB and decide whether R is moving forwards or backwards in time relative to the cosmos.

This is basically the resolution to (most) time-travel dilemmas in the initial-value/Hamiltonian formulations of General Relativity: equip the spacetime with a distribution of matter (the cosmic microwaves) and nonzero cosmological constant (cosmic expansion or contraction) and use those two features to define a cosmological scale factor that one can use as a principled clock.

Physical cosmology is some complications on this picture: we have other fluids as well as the CMB, and they dilute away (in the direction of expansion) differently.

There are also of course observers who are moving extremely compared to the slowly-separating galaxy clusters, not just relativistically but at enormous accelerations. These observers are a tiny fraction compared to the cosmological observers.

(A bigger problem is that the clumpy observers -- galaxy clusters and things in them -- have to be "stitched in" to the cosmological frame, and that must be a mass-dependent calculation, and is dealt with in various ways including "swiss cheese" models or using thin-shell Israel-Darmois junctions. However the corrections are small and the thermodyanmic arrows of time are still very closely aligned, so occupants of galaxies, even near massive black holes, would still prefer to think of t_n as picking out the difference between forward-time-travelling R and backwards-time-travelling-R).

Finally, it is in isolation that the dynamical evolution of R (and e) shows serious causality or closed-system-thermodynamics violations. We would expect to repair that by embedding R (and e) into the wider foliation, which is built out of a block-universe equipped with several spacetime-filling fields. If the block universe contains a pattern like the second diagram above, then the dynamical field equations must obviously allow them. But Geroch's point is the other side if this: let's start with the known dynamical field equations, and see where they take us. They might not take us to a schematic like the one above, and the absence of observations like the picture above weighs heavily towards laying down some initial data (say at t_-1000) and seeing how it evolves. Maybe it can evolve to show t_0 through t_5, maybe it cannot.

Relatedly, https://en.wikipedia.org/wiki/One-electron_universe where instead of flipping the thermodynamic arrow of time of our backwards-and-forwards time traveller, we are flipping the sign of the electric charge.