Maybe you can show me where in any of the things he wrote he says anything like what you claim.
Perhaps I can! However, do you know who coined the phrase - "genius is knowing how to hide your sources"?
How can something that appears to happen in nature conflict with nature?
I agree, this is a somewhat nonsensical question. There's nothing you can say that can't be said : there's nothing that can happen in nature that conflicts with nature.
Something can appear to happen that is in conflict with nature (the limited slice of it we know) when it only APPEARS to or our understanding of nature is wrong/incomplete.
And it wasn’t the “action at a distance” that conflicted with relativity. It was the instantaneous action at a distance of Newton’s gravity.
Both are a major problem for gravitation. As I said, this is something that most all physicists since newton understood. By einsteins time it was a famous problem in physics, almost centuries old by that point.
SR showed that nothing can propagate through space faster than the speed of light, including gravity.
This would be ONE major problem for instantaneous action at a distance, yes. The major one we are discussing, however, is the acute lack of mechanism/description/definition of WHAT gravitation is and how it accomplishes its many miracles.
Newton assumed gravity acted instantaneously, so relativity was incompatible with that. Relativity solved that by showing that gravity doesn't act instantaneously.
If by "solved" you mean "ignored" and "showing", you mean "assuming" - then yes, that's correct.
That’s true. The challenge is coming up with a theory of cause is that it is consistent with the effects we perceive.
Gravitation IS the theory of cause. It's just piss poor.
Relativity is an aether theory (though I doubt you were taught this). It mathematically presumes a physical substrate to "empty" reality which contorts with the presence of mass. The contorting of this aether is the presumed cause of gravity in relativity. Empiricism demands that we provide observational and experimental support for the theoretical entity itself (aether/spacetime) as well as the mechanism responsible for matters contortion of it from a distance. Because we excised aether from the schools, and physics is so poorly taught generally, there was believed to be nothing to go looking for (in part because many that tried found nothing).
the spacetime curvature theory meets all the criteria for explaining how bodies (and light) move.
Not mathematically speaking. In any case, we are talking about gravitation. "Spacetime curvature" is just more meaningless gooblety-gook. Nothing is proposed (or known) to do the warping, nothing is proposed for how it does so at an infinite distance, nothing (mostly) is proposed to be within the two gravitating bodies. In newtonian theory, something (matter) acts upon nothing (space) to affect the motion of incredibly distant matter (something). Reduced/generalized this is : Something acts upon nothing which acts upon something. This is fundamentally unacceptable in physics and also what relativity sought to change with its mathematical description of gravitation.
GR predicts motion of bodies and light perfectly, at least so far.
Also not true, but even if it were - this is a discussion about gravitation NOT relativity.
Again, we have been able to measure those geometric differences and they are consistent with the field equations.
You can't measure something that doesn't exist (or isn't even defined well enough to know how/where to look anyway).
http://www.thephysicsmill.com/2015/12/27/measuring-the-curvature-of-spacetime-with-the-geodetic-effect/
I will check this out. I don't think I've encountered "the geodetic effect" before.
Spacetime curvature can also be measured by testing how the gravitational field changes as over distance.
So you believe, and you MUST believe - otherwise you can't continue to measure weight and BELIEVE that an invisible undefined "magic" of gravitation is behind the scenes causing it.
The difference in the measurements can tell you how much it curves.
And ASSUME how much it curves, IF it were real, curvable, and being curved by "SOMETHING". Do you see why this is something very different from empirical science?