Is there a natural coordinate system in general relativity? After reviewing some different coordinates systems that describe Schwarzschild spacetime (such as Gullstrand–Painlevé coordinates) it seems like we can always make a coordinates transformation very far from us such that locally we are in Minkowski coordinates.
So the question if whether one of those coordinates will be "more natural" to work in (so are there useful general condition that give rise to a unique coordinates).
Of course that locally all the laws of physics will not prefer any system, but in this condition I allow also extrinsic experiments (similarly to the uniqueness (up to translations and rotations) of the coordinates used in the Friedman's metric that follows when we require Homogenic and Isotropic coordinates)
Edit:
If I measure different velocities to the same object that is significant difference for me. The fact that you can deduce the same conclusion from experiment only mean that that every measurements correspond to some invariant property. Saying that there is no general conditions that we can define in order to get always some defined properties doesn't follow from that. Note that this is not a formal mathematical question but rather a question on what we can define if we want to and there is always a possibility that after we define something it will turn out to be natural to most of us
It seems to me like there are classes of equivalence that we can define that have some meaning. In this case I mean that we can define all the coordinates transformation that does not involve time to be in the same class and then for example cartesian and spherical be in the same class. And I argue that this class is unique in some sense and I am trying to formulate specific (but more general) condition for other systems.
I want to create the uniqueness in some sense that is meaningful to me using those terms and I wonder what such condition will look like
 A: Coordinates literally make no difference in any physical sense. Is there any sense in which polar coordinates in the plane are a more ‘true’ description of (an open subset of) the plane? Absolute not. Thus, there is no natural/ inherently preferred coordinate system.
A completely separate question is whether there are situations where we might favour one set of coordinates over another. For that the answer is yes. The choice of coordinates depends entirely on the problem at hand. Some coordinates may make some features of geometry more apparent, for instance:

*

*in the usual Schwarzschild coordinates $(t,r,\theta,\phi)$, the metric components do no depend on $t$ or $\phi$ so $\frac{\partial}{\partial t}$ and $\frac{\partial}{\partial \phi}$ are Killing fields, whereas in Kruskal coordinates $(T,R,\theta,\phi)$, the metric components do depend on $T,R$, so $\frac{\partial}{\partial T}$ is no longer a Killing field.

*if you’re interested in studying things like propagation of waves, or just null curves, then it is a good idea to not use the usual $t,r$ coordinates, but replace them with some form of $u,v$ null-coordinates which will more transparently reflect the geometry (e.g in Minkowksi, we use the two $45^{\circ}$ rotated lightlike coordinates instead of the usual $t,x$).

Anyway, this is not at all a feature specific to GR. Even in classical mechanics, the coordinate system we use is the one which is most conveniently adapted to the problem at hand (rotation: use polar coordinates, translations: use cartesian, for a mix of both, make a choice as to which aspect you wish to simplify etc).
A: The natural coordinates don't depend so much on the theory as on the experiments you perform. The phenomena, not the models, are fundamental to physics.
