Dear Richard J Benish,
As you note elsewhere, we both have high regard for Tom Phipps' contributions to physics, despite certain disagreements with his approach. You further point out something I believe often goes unnoticed:
"...understanding a theory about gravity (i.e. GR) is often confused for understanding the physical phenomenon of gravity itself."
As you say with reference to "matter tells space time how to curve, and space-time tells matter how to move", no academic physicist bothers to point out that we have no idea how these orders are carried out. You extend this line of criteria to "quantum gravity", and to how "gravitons" work, in that they make no physical sense. Just part of quantum field theorists attempt to force the universe into a bookkeeping scheme.
With respect to your comments above, your first paragraphs effectively summarize the situation. I agree that one-way measurements are hard, perhaps impossible, hence the average back-and-fourth measurements predominate. I have designed an experiment that should be capable of measuring the velocity of the local frame from within the context of the local frame with no outside information. This should establish whether my approach is valid or invalid.
The experiment you discuss has never been done, yet, like other 'gedanken' experiments, it is typically accepted as reality. It's not quite clear to me why achievable experiments that question the status quo are not performed. I hope both of our experiments will be performed.
You then discuss maximum geodesics and accelerometers. My own perspective is that "curved space-time" outside matter is equivalent to energy density distributions in flat space. As you probably know, Weinberg, Feynman, and others have shown that iterated flat space approaches lead to Einstein's field equations in "curved space" so my inclination is to reject "curved space-time" (incapable of dealing with "density" or with "self-interaction energy") and this bias extends to rejecting higher dimensional theories of physics. You identify the motion as not through space, but of space, and view this as curvature in (4+1)D. My perspective on the gravito-magnetic ('C') field is analogous to electro-magnetic circulation, i.e., circulation of the field with characteristic angular momentum. Circa 2006 Martin Tajmar used accelerometers to measure gravito-magnetic field circulation. I reject higher dimensions of space, from 4 to 11, however it might be possible to interpret circulation in space as a fourth dimension. This is more a mathematical representation, like the Minkowski representation, than a true description of the physics. Clearly n-dimensional representations are of utility in physics. Having read your essay several times I'm still not exactly clear on how your (4+1)D model is to be interpreted. My 3D mind, operating in time, works well with n-dimensional math, but does not grasp spatial models greater than 3D.
As for the accelerometer questions (ignoring gravito-magnetic issues) it is probably not purple-winged horsies, but the gravity gradient dG/dt that imparts momentum and induces local gravito-magnetic circulation. How this registers or not on an accelerometer is not clear to me, having not studied accelerometers in ages. The equivalence principle that falling 'cancels' gravity, does not prevent the accumulation of kinetic energy.
In summary, I do not intuitively grasp how a gravitation field 'pulls' and I don't think 'gravitions' is the answer, nor do I accept 'curved space' explains anything physical. 'Pushing' seems to bring with it another set of problems, and might work for a universe with only one central body, but I can't envision a many-body dynamics in such a case. Gravity to me is the great mystery, acceptance of which seems to unlock other doors big time. Neither gravitons, curved space-time, 'dynamic space' nor (4+1)D do it for me, yet I feel the field as I just sit here typing. It's real, and when I accept the reality, and play with the equations, lots of the universe falls out. I know this doesn't answer your questions, but it's a mystery to me.
I appreciate your many comments. If we ever meet, let's drink to Tom Phipps.
Edwin Eugene Klingman