First, to say, " I reject a physical theory (GR) which predicts everything that we can currently observe about gravity and the nature of spacetime in favour of one (Newtonian mechanics) which demonstrably fails to predict some observations correctly and which is therefore a less accurate description of reality " is not my position at all
Indeed it is your position - the theory is a description of reality. By rejecting the foundation of GR which is that spacetime curvature is what gives rise to gravitational effects, you are de facto rejecting the theory in favour of one in which space is universally Euclidean and which makes poorer predictions - ie is a less accurate description of reality.
As regards the theory of ( GR ) I make no more judgment than scientists make, it seems to work. So I do not reject the theory, I reject its interpretation - that space and time curve and that photons have no mass.
Since GR is based on the concept that spacetime is curved by the presence of mass-energy, your claim that you accept the theory but reject its interprtetation is a distinction without a difference.
Hecd2 has made a is a value judgment without merit. Are we to think then that the " bending of space, the bending of time " are applicable only to light?
Of course not - why would we think that? The Einstein field equations which embody the concept of spacetime curvature predict everything that Newtonian mechanics and Special Relativity predicted and more - so is more accurate under certain conditions and this applies to all bodies.
But if it applies to all objects in space, then it would seem that every solar system would have its own pecular system of curved space and time. Would this mean that all curved space-time is purely local?
Exactly so, spacetime is locally curved by the influence of all massive bodies in the universe, but the influence of nearby bodies is greatest. The curvature varies from place to place.
As Lord Whimsy might say, " Very odd indeed. "
What Lord Whimsy says or thinks about this is entirely irrelevant to whether it is so or not.
And wouldn’t there be, somewhere in the universe, conflicting paths of this curved space-time, as between solar systems or between galaxies?
I don’t know what you mean by paths, but GR predicts that far away from massive bodies spacetime tends to the average geometry of the Universe, which has been measured to be flat or nearly so.
Under Hecd2’s explanation, we would have to discard the notion that gravitational bodies, such as any object larger than a photon, are governed not by the attraction and repelling of gravitaty but by the bending of space-time!!
It’s not my explanation but that of General Relativity. But the curvature of spacetime is an explanation (which is correct as far as it has been measured) for why we observe the attraction of gravity (gravity doesn’t repel).
It is more consistent with the nature of things to recognize that we can have a mathematical formula which gives the correct outcome but does not reflect the nature of things.
Only if you insist that your preconceived intuition is right in the face of contrary evidence.
So it is more logical to say that nature always remains the same, irregardless of a particular mathematical formula, that space does not bend and that time does not bend and that photons have mass like every other body in space, and that the mass of photons is overcome by the gravitational mass of other greater gravitational masses?
No, and the idea that one mass is “overcome” by another is bizarrely wrong, even according to Newton.
Furthermore, are space and time real substances like a stick or a fishing rod that can bend?
The foundational idea of GR is that space is a three dimensional manifold (like a sphere is a two dimensional manifold), which can be flat (locally parallel lines are always the same distance apart and the sum of the angles of a triangle add up to 180 degrees and so on), or not flat (locally parallel lines can meet or diverge, angles of a triangle are more or less than 180 degrees etc, as they are on a sphere in the two dimesnsional example) and that the curvature of spacetime is determined by the presence of mass-energy.
And did Einstein actually say that space curves and that time curves? If it wasn’t him, who first made that philosophical interpretation?
It’s not a philosphical interpretation - it’s built into the foundation of the theory in the Einstein Field Equations which is where he said it first.