Well, yes, this is just ONE of several ways in which the concept of 'locality' no longer makes sense.
I think you are using that term "locality" in a different way than I would normally use it.
Even in the purely classical system of SR/GR the idea of 'here' and 'there' don't actually add up.
Well, that is probably only because "don't add up" is not a well-defined phrase, so that nobody outside your head knows what you mean.
If to "add up" they must be objective absolutes, then yes, they don't add up. But your experience of the color blue is not an objective absolute, either, but you and others can talk about it quite sensibly.
At best there are 'relations' between things, which provide some localized basis from which to build a coordinate system, but in GR you certainly can't extend it.
Extend it in what sense? And "can't" in what sense?
I mean, consider that even without GR, there are more coordinate systems that are problematic than not. Geocentric coordinates were the first we ever used, even before Newton, and those coordinates *sucked* for just about everything other than talking about the Moon. You'd not want to extend that to infinity and try to describe movements in the universe in it, even if spacetime didn't curve around masses. It isn't so much "can't" as "Nobody sane would want to."
Or in the case of relativistic quantum field theory one can wonder if they really tell you much of anything at all, since somehow that photon interacts with that electron no mater what.
It is perhaps more accurate to say that time and space are still fine, the problem is that the photon never actually exists, but we insist on using it to describe the interaction anyway. That is not spacetime's fault.
