I'm sorry, no, this is even less clear to me.

GR has coordinate freedom built into it; you can apply (almost) any set of coordinates (locally) on spacetime and calculate against them as you wish. The metric, however, is fixed by the geometry itself, and it is the metric that determines the interval in any set of coordinates. That's why you can write down e.g. the Minkowski metric in cartesian or spherical coordinates and come up with the same dS^2 -- the spacetime interval -- or as another example the Schwarzschild metric in Schwarzschild coordinates, in- or outgoing Eddington-Finkelstein coordinates, Kruskal-Szekeres coordinates, etc etc and come up with the same dS^2 even though the line element is written very differently in each case, and even though coordinate singularities appear in some systems of coordinates but not in others. (That they vanish in any valid coordinate system shows that they aren't physical singularities; the singularity in a Schwarzschild black hole that does not vanish under at least valid coordinate system is a strong argument that it is physical, although that remains a subject of research.)

One of the main features of GR is that it gives you a clear and mathematically provable mechanism for translating from one set of valid coordinates to another. That's one of the main reason it's called General : the coordinate systems need not be in uniform motion with respect to one another, and even more crucially, they do not even have to be in the same tangent space. There is nothing at all arbitrary about how one "transfers" from one system of coordinates to another: the Bianchi identities on the Einstein Field Equations give the degrees of freedom necessary for a gauge transformation.

The physicality of singularities in some exact solutions of the Einstein Field Equation (like Schwarzschild's as mentioned above) and the work of Kenneth G Wilson has led most general relativists to think of GR as an effective field theory (EFT) valid everywhere just outside of irremovable singularities. That means GR is accurate even inside the event horizons of black holes. Really only the singularities themselves are the problems with GR -- in particular it's the problem of its canonical quantization, and thus the driver for a theory of quantum gravity that works much closer to the irremovable singularities (or better still, makes them vanish).

But because GR is easy to consider an EFT, it is even easier to treat regions of spacetime that are not exactly flat as if they are exactly flat with some perturbations. So, in fact, I do not at all agree with you that "... you have to take into account interaction with many planets and other celestial bodies ..." since in a perturbative approach, their contributions can be removed as irrelevant. Even nearby celestial bodies can be removed as marginal. Here "marginal" and "irrelevant" are vocabulary right from Wilson, and translate very nicely into a Feynman diagram approach by counting loops of gravitons.

Or alternatively, I could technically agree that you have to take into account those bodies you mention, but since you do so simply by ignoring them explicitly on the grounds that their contributions to the metric provably do not matter for the local system under consideration, that contradicts your assertion in the paragraph "Similarly...".

I don't understand how your fourth last to second last paragraphs inter-relate. The first of those approaches an argument about coordinate freedom; the second I cannot decipher at all (it certainly isn't a General Relativity argument; I think you're looking to argue with the Standard Model instead, and that's defined on flat spacetime, and by the EFT argument above, flat spacetime applies in any radiocarbon dating experiment you choose to do on Earth -- and when you repeat the same experiment, the secular variation in the proximity of the the moon or sun or other planets will make not even a marginal difference). And your second last paragraph is I think a question about multimetric gravity -- we have strong evidence for the universal coupling of everything in the Standard Model to the single metric, and if you want to introduce a second metric for some sector of matter, you have to go beyond the Standard Model, and you have the problem of making the coupling just universal enough that you reproduce GR's matches with observation. There are ways of doing that, but most of them have the extra metrics decay extremely early in the history of the universe so that they do not produce ANY observables. Afshordi and Magueijo's recent paper is interesting because they believe they can produce an observable in spite of a decaying extra metric to which light couples. That comes at a cost though: you would have to introduce a zoo of gauge particles in order to make this work at all, which hardly makes the theory more simple than Cosmic Inflation.