Why is it said that the physics is preserved when the equations of motion are invariant under a tranformation/Symmetry? It is always said (and this is even how physicists motivate some theories instead of others) that the physics is preserved (by some transformation) whenever the equations of motion are left unchanged by some transformation and/or symmetry.
This yet seems completely false. In General relativity for example, because Einstein's equations are tensorial equations, their form are fixed once and for all regardless of the coordinate chart you chose to look at them from. The thing is, different coordinate charts will obviously give different physics.
If you chose, say, a geodesic frame, by very definition the Christoffel symbols are zero there, which means that there is no acceleration (which is the definition of inertiality). This is obviously wrong in a "generic" frame.
It seems to me that you can always trick syntactically to preserve the form of the equations of motion, even in newtonian mechanics: just add all of the fictitious forces on the left side of F = ma, and make it such that in an inertial frame those are exactly zero. Yet, the physics is obviously very different in an inertial frame than what it is in an accelerated one (as in, the true physics claims in one frame are obviously not true in another).
So my question is simple: how to know what does preserving the equations of motion by some transformation tells us about the physics? As in, what physics claims are preserved when the equations of motion are preserved by some transformation?
 A: Symmetries are deeply connected to physical laws, and most fundamental laws like Energy, Momentum, and Angular Momentum conservations are the consequences of symmetries. You must be knowing that the laws of Physics are described using the Lagrangian formulation. 
When Special Relativity came, it made the point even stronger, by claiming that the laws of Physics are the same in all reference frame showing some form of a Relativistic symmetry.
Then we come to the point about General Relativity. You have pointed out that when we switch reference frame we get different equations of motion. I would like to point out that this is because you are observing the same event. An analogy to this may be considered as follows. You are on the ground, and you throw a ball up, it comes back to your hand. Now someone in a car moving at a constant speed throws a ball up, and it comes back to his hand. His description of the motion of the ball is the same as what you measured for your ball, but your description of his ball is obviously not going to be the same as your description of your ball. Same goes with the equations of the motion in General Relativity. But what it says is that the Tensor equations, which are the laws of motion require no modifications, just like Newton's laws needed no modification for you or the person in the car, when both of you were observing your respective balls.
A: Suppose you assume the opposite: that physics predicted by the equations of motion becomes different if I change the equations of motion under a symmetry they obey. That does not make sense: by definition, if the equations are symmetric they are unchanged by the symmetry, and must make the same prediction.
For example, consider Newtonian mechanics under translation. We can write the equations in a vector form that is independent of (Cartesian) coordinates, just like the Einstein equations are independent of coordinates more generally. I can select centre of mass frames or co-moving frames in Newtonian mechanics, making a particle move or not relative to the frame. That does not change the underlying physics! It is just that different observers would see different things, just like you and me disagree on whether an object is to the left or right when standing face to face.
If a different coordinate chart gives you different physics in GR, then you have made a mistake somewhere. Either that the chart actually represents a different spacetime, or that you are assuming the two different outcomes are what the same observer could see (nearly all "paradoxes" in relativity consist of getting confused about this).
Levels of physical theorizing
What I think is going on here is that one needs to distinguish between (1) the actual world with whatever physics it actually has, (2) the exhibited physics in a good model (like Newton's or Einstein's) that closely fits the actual world, (3) the mathematical framework the model uses to make predictions, and (4) the conceptual model itself.
The first paragraph of the answer is just stating that symmetry on level 4 implies symmetry on level 3 and 2. When I make my calculations on level 3 I can use all sorts of shortcuts, cleverly selected frames, or approximations to say what happens on level 2 - if I miscalculate, or use the wrong representations I can get discrepancies with level 4, but that is my problem, not a problem for the theory or reality. When things work as they should, all approaches on level 3 should give the same measurable outcomes on level 2 (e.g. angles of deflection, interaction probabilities etc.). Meanwhile, what makes us want to use the theory is that these measurable predictions on level 2 fit very well with measures made on level 1.
But it is not just individual predictions that should fit, but entire classes: the more a succinct theory explains large amounts of data closely the more we think it is a good theory. Symmetries are bold predictions predicting that measurables are invariant under large classes of transformations that are also tightly linked to the structure of the theory. That makes the theory more falsifiable, and the ghost of Karl Popper will not haunt you.
A: Your point is discussed at length, for example, in

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*Fock: The Theory of Space, Time and Gravitation (Oxford 2nd ed. 1964),

and also in

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*Synge: Relativity: The General Theory (North-Holland 1960).

It is indeed true that one can reformulate Newtonian mechanics in a completely invariant, "space-time" way. Such a reformulation is called (among other names) "Newton-Cartan theory", because of Cartan's role in its development:

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*Cartan: Sur les variétés à connexion affine et la théorie de la relativité généralisée, Annales scientifiques de l'École Normale Supérieure 40 (1923), 41 (1924), 42 (1925).

See for example

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*Earman, Friedman: The Meaning and Status of Newton's Law of Inertia and the Nature of Gravitational Forces, Phil. Sci. 40 (1973)

or

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*Ellis: Relativistic Cosmology, pp. 104–182 in Sachs (ed.): General Relativity and Cosmology (Academic Press 1971),

which develops Newtonian mechanics and general relativity literally side by side, to explain the difference in their spacetime-geometric structures.
The difference between them is in the role of the affine connection and the 4-metric in their equations. In particular, Newton's "spacetime equations" lead to the selection of a special family of spacelike slices (the one associated with "absolute time"), and the evolution of its 4-metric isn't influenced by matter fields. Einstein's equations lead instead to the selection of a special family of lightlike hypersurfaces, and the evolution of their 4-metric is influenced by matter fields.
This is why Fock stated that the question of "general invariance" isn't really a question of physical symmetry. Although we can discuss about the symmetries of specific solutions of general-invariant equations.
Sometimes a physical theory is described from a meta-theoretical point of view, and some of these points of view may include some sort of physical or "protophysical" notions. It is often subtle to distinguish what is in the theory and what in the metatheory, and such division is somewhat arbitrary. One must be careful and precise in one's definitions and assumptions. Works on these matters are, for example:

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*Bunge: Laws of Physical Laws, Am. J. Phys. 29 (1961)


*Bunge: Foundations of Physics (Springer 1967)
and of course Poincaré's writings, collected for example in

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*Poincaré: The Foundations of Science (Science Press 1946)

and Duhem's:

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*Duhem: The Aim and Structure of Physical Theory (Princeton 1991).


Regarding "the physics of this or of that", I think that's a very vague term, sometimes used to sound cool. When it has any meaning it just means "the equations that describe this or that".
