I don't disagree with your overarching observations, but it's not so much that there are "pieces" of General Relativity to examine, but rather that the most central two features of the theory produce -- inevitably -- several consequences, all of which seem to be supported by observational and experimental evidence. Adapting the central two features generically leads to a universe noticeably different from what we observe. Likewise, the universe we observe appears to demand General Relativity as a framework for describing it accurately.

Although there is some history about which empirical results were elevated into postulates of General Relativity, the history of the theory's discovery and development is pretty irrelevant to the two key features of General Relativity as a theory for our universe. They are: the Einstein tensor of curvature is constantly proportional to the stress-energy-momentum tensor of matter, and the universe's spacetime is a Lorentzian manifold with a vanishing torsion tensor. These two theoretical features produce dramatic consequences which are physically testable.

The two basic avenues of attack of General Relativity as a theory accurately describing our universe: undermine local Lorentz invariance (if it's wrong then we aren't in a torsion-free 4-dimensional pseudo-Riemannian space), or undermine the strong equivalence principle (specifically the universality of free-fall, which means that inertial mass and (passive) gravitational mass are always identical) which would blow apart the proportionality of curvature to matter.

The first line of attack tests the consequence that locally -- within a laboratory, say, whether that's on the ground, on the ISS, or on the moon -- physics which aren't dependent on gravitation are compatible with Special Relativity. Since the Standard Model of Particle Physics is governed by Special Relativity and is extremely sensitive to deviations from it, and is totally silent on the subject of gravitation, tests of the Standard Model of Particle Physics are also probes of this consequence, which is called local Lorentz invariance. Proof that results of Standard Model experiments differ during odd minutes from those obtained during even minutes, or that such experiments produce different results when the ISS is over the northern hemisphere rather than over the southern hemisphere, would call into question universal local Lorentz invariance. These tests happen often, and a lot of attentionis generated whenever a claim is made which violates local Lorentz invariance. Practically all such claims have been found to have been wrong. https://en.wikipedia.org/wiki/Modern_searches_for_Lorentz_vi...

The second avenue is to look for cases where black holes or neutron stars travel on different orbits from ordinary stars, planets, asteroids, people, or flecks of paint falling off spacecraft. Observational evidence favours this part of General Relativity, which implies that the gravitational constant G is truly a universal constant (the same everywhere and at all times and in all systems of matter), and that there is no long range interaction that applies to all mass-energy including the Standard Model and whatever dark matter is. 8 pi G (where c is set to 1) is the constant of proportionality between the tensors mentioned above, and it's not General Relativity if that constant is not the same at all times and in all places. (This exact proportionality also protects the components of the Einstein curvature tensor: it's not General Relativity if you allow the Riemann scalar ("R") to vary, and there are plenty of alternatives which propose to vary R: these are f(R) gravity theories and most conflict pretty violently with what we observe in the universe (technically General Relativity is an f(R) gravity where f is the identity function).

These two postulates of General Relativity produce numerous consequences which are unavoidable, and so there is a wide range of tests of General Relativity that are implicitly tests of these two postulates. These consequences can be described parametrically in such a way that alternatives to General Relativity can have their consequences assessed: deviations from General Relativity in most of these parameters is strongly constrained by observation and experiment, and it is maddeningly difficult to produce an alternative to GR that survives this kind of theoretical analysis. It's easy to shoot down most f(R), f(G), or f(c) theories using a parameterized Taylor-like expansion, for example. It's a pity, because a number of such theories would significantly simplify the description of the hot dense phase of the universe about 13.8 billion years in our past.

Worse, so far nobody has found a parameter extracted from General Relativity which is contradicted by observational or experimental evidence, and most tests of such parameters are in accord with General Relativity to better than a fraction of a percent, and in many in some cases to much better than a part in a billion (we have some statistics from the Hulse-Taylor system that constrain to within one part in 10^20 some preferred-frame effects that touch on local Lorentz invariance and the universality of free fall).

The problem is that General Relativity itself is mathematically complete and the maths say that gravitational interactions are entirely dependent on the distribution of matter (in the most general sense), and there's lots of detail we don't know about how matter is distributed or how exactly a lot of it works, including in the Standard Model at extremely high energies and also at extremely low energies (e.g. justifying infrared cutoffs on soft photons). Modifying General Relativity as an approach to make extreme-energy particle physics simpler has tended to produce a lot of dead ends.