This issue is discussed somewhat in Steven Weinstein's essay "Naïve Quantum Gravity", published in Physics Meets Philosophy at the Planck Scale: Contemporary Theories in Quantum Gravity (eds. Callender & Huggett, 2001). He begins by noting that
[a]n alternative way to conceive of gravity would of course be to follow the lead of other theories, and regard the gravitational field as simply a distribution of properties (the field strengths) in flat spacetime.* What ultimately makes this unattractive is that the distinctive properties of this spacetime would be completely unobservable, because all matter and fields gravitate. In particular, light rays would not lie on the "light cone" in a flat spacetime, once one incorporated the influence of gravity. It was ultimately the unobservability of the initial structure of Minkowski spacetime that led Einstein to eliminate it from his theory of gravitation and embrace the geometric approach.
So it's perhaps more elegant to view GR as a theory of curved spacetime; but could we get away with thinking about it as a weird non-linear field theory on Minkowski spacetime anyhow? It's certainly possible in some circumstances; famously, Steven Weinberg's 1972 book on general relativity tries to eschew geometric thinking as much as possible, viewing GR as a field theory that has the Equivalence Principle as a fundamental principle and showing how object such as the metric and curvature tensors can be thought of as arising from this principle. It seems to me that this is not far off from the OP's idea in the second paragraph that "any set of assertions compatible with the prediction of GR as we understand them today (again, when expressed in a GR-free and purely physical language including background independence) necessarily leads to GR and the identification of gravity with spacetime", though Weinberg (in 1972) might have disputed the last part.
However, Weinstein notes that there are still a few problems with this approach:
First, the "invisibility" of the flat spacetime means that there is no privileged way to decompose a given curved spacetime into a flat background and a curved perturbation about that background. Though this non-uniqueness is not particularly problematic for the classical theory, it is quite problematic for the quantum theory, because different ways of decomposing the geometry (and thus retrieving a flat background geometry) yield different quantum theories. Second, not all topologies admit a flat metric, and therefore space times formulated on such apologies do not admit a decomposition into flat metric and curved perturbation. Third, it is not clear a priori that, in seeking to make a decomposition in the background and perturbations about the background, that the background should be flat. For example, why not use a background of constant curvature?
* Weinstein includes a footnote here which refers the reader to an "interesting philosophical analysis of this line of thinking" in Reichenbach's The Philosophy of Space and Time (Eng. translation), (1958 [1927]).
