I just noticed this:
> our rotation causes frame dragging
Not exactly; it's the choice of accelerated coordinates and pretending that the coordinates are freely-falling that manifests seemingly odd coordinate-dependent physical effects. One can resolve these by switching to freely-falling coordinates, or by not pretending that the accelerated coordinates are freely-falling. In practice this means doing Special Relativity calculations only in the Special background of the theory, and using the full covariant physics otherwise.
One can treat the Lense-Thirring effect as a distortion of a Special Relativistic background (or a different static background, like Schwarzschild), or one can determine the actual background from the distribution of stress-energy.
The first choice leads one to conclusions like the precession of test objects in polar orbits around axisymmetric bodies with nonzero angular momentum, and applying that to Earth and a satellite.
The second choice is more work, because angular momentum and axisymmetry are only two of the departures from Schwarzschild in the system. Crucially the contributions from surface bumps, masscons, the distribution of matter in the satellite, and so on are all small enough that it's fair to ignore them in the second case.
The difference is that the second case simply correctly calculates out the geodesic the satellite occupies in the real spacetime, while the latter calculates out an orbit that is simply wrong for the real configuration of the system and then applies corrections by bringing in pseudo-forces.
In the weak field limit, one can get completely correct coordinate-dependent results taking the former approach. Additionally, analogies arise in this approach that can lead to interesting intuitions. However, one should always check to make sure that the intuitions can be explained in terms of coordinate-independent formulations of physics.
> weird Sun-Earth orbital relativistic effects
If you calculate out the geodesics, no; all the parts of Earth down to its individual molecules and below "want" to travel on geodesics sourced by the system and do so unless the stronger three forces interfere with that "want" (and in bulk Beiglböck and Dixon show that the Earth has a coordinate-independent centre-of-mass that does travel on a timelike geodesic, and you arrive at it by considering the vector position of each particle).
This is a lot of work.
So instead, you can use a background like Kerr (or Schwarzschild or Minkowski) and correct the failure of the parts of the Earth (or the planet in the large) to travel on the geodesics of these backgrounds. The correction is typically done by introducing pseudoforces, pseudofields, and dynamics and potentials in these. They're pseudo because they vanish entirely in at least one frame of reference. (see https://en.wikipedia.org/wiki/Fictitious_force which is pretty decent).
> gravitational aether
One could do perturbative General Relativity by choosing Minkowski (flat) spacetime as the background \eta [0], and then recovering all the real motions of objects in a perturbation field h. That field, h, is a pseudofield with its own potentials and dynamics. It's just a set of first-order corrections to the background metric. The real metric g will be an expansion: g = \eta + h + O(h^2) + O(h^3) + ... where O(h^x) are higher-order correcting terms, and those are negligible for systems where stresses are weak and speeds are slow compared to c.
It is reasonable to think of h as having the properties of an aether in some cases: it can walk and quack like a real field, like the electromagnetic one in Maxwell's theory. However it can also be made to vanish entirely without removing matter and the gravitational potentials they source, and it is a pseudofield in the more fundamental theory of General Relativity. The Maxwell electromagnetic field is always present (you can only get rid of it by removing all charges and potentials), and obviously related fields are still there in the more fundamental theories of QED and the Standard Model (a QFT).
The specific theory of the luminiferous aether theory that Michelson-Morley surprised themselves by falsifying was that the aether was stationary and the Earth moved through it. There were comparable theories of a gravitational aether (https://en.wikipedia.org/wiki/Mechanical_explanations_of_gra...) that have never recovered many fairly easily observed behaviours of gravitating masses, so there was never really a surprisingly failed direct test of them.
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[0] More generally, as someone raised this with me before, one can write g^{(0)}_{\mu\nu} for an arbitrary choice of background instead of abusing the notation \eta_{\mu\nu} which (out of context) generally means only the Minkowski metric. I leave out the greek-lettered indices above.