> preposterous

Well, specifically, proper-time intervals are path-local in General Relativity (GR). There is a unique coordinate-invariant proper-time interval between two points on a timelike worldline.

I think that extending that to "time is a localized phenomenon" is harder than it seems, notably because worldlines depend on the full solution of the Einstein Field Equations. In a Big Bang cosmology (with a hyperbolization of the EFEs and ignoring constraints and diffeomorphism freedom), it's pretty brave to deny a relationship between the early boundary and the values of the fields at any point p on the manifold given that the causal cone at p of M contains the Big Bang.

> effect that local gravitational force has on the passage of time

It's the metric that leads to Lorentzian observables between observers at different points in the manifold. The metric near bodies like Earth closely approximates that of Schwarzschild spacetime in the way it generates geodesics including the null geodesics (among others) carrying information from one observer to another. Effects like gravitational redshift arise from the fact that in spacetime more-curved paths are shorter than less-curved paths (as opposed to how curved paths are longer than straight paths through Euclidean space).

The metric's generation of geodesics is difficult to relate to a classical force or potential in general. Two objects in vacuum free-fall can be at different gravitational potential while feeling no force whatsoever; the one at higher potential ticks faster. It's a bit easier in near-Schwarzschild. Consider two atomic clocks falling from different altitudes[1] towards the same point on the (practically atmosphere-free) moon; almost all observers will agree that the higher clock runs faster than the lower clock until they are both smashed together on the surface. Yet if each clock is equipped with an vector accelerometer, both accelerometers will point nowhere in particular with a magnitude of zero from the start of their free-fall trajectory until collision with the moon's surface -- the first time force is reported by the accelerometers is when "lithobraking" starts.

However, properly considering gravitational potential as a 4-vector generally requires some choices which eat the redundancies in the Einstein Field Equations. In General Relativity one has only the metric and Christoffel symbols and tedious arguments about which mathematical objects correspond to a Newtonian notion of a gravitational field (answer: "it depends" or "none of them"). Gauge-fixing lets one set a "depends" condition such that one can recover a vector potential field and a scalar field strength at each point; this approach is taken very seriously in Fedosin's covariant theory of gravitation for instance.

> effect that local gravitational force has on passage of time

Even if one takes steps to model some aspects of the gravitational interaction as a force, the proper time interval of an object doesn't change with the force acting on it. But the frequencies, lengths and related quantities of an object at some distance does depend on the force the object feels compared to the force the observer feels. (Moreover, if observers are in vacuum free-fall then they will feel no force at all, and can only infer the gravitational interaction from either a deviation from a straight-line track on a choice of coordinates, or by comparing the ticking rates of their own wristwatch with the wristwatch of several observer at some distance -- from [Synge 1960] this would take a minimum of five freely-falling wristwatches in total).

Generally the complexities of setting down this kind of gauge-and-coordinate conditions leads relativists away from worrying about relating GR's mathematical objects and Newton's, and it's easier to say "gravitation is not a force" rather than "with some effort you can treat gravitation as a force in local coordinates and in a local gauge but you'll still find yourself returning to the Special Relativistic forms of physics equations because they genuinely are the simplest form and are always valid in the neighbourhood around a point on a geodesic".

> if one considers the theories and research related to what I linked above

General Relativity is in extremely precise accord with observation at many length scales and
direct experiment within the solar system. Deviations from General Relativity that are different in the limit of the parameterized post-Newtonian formalism (which applies at solar system scales) are almost entirely ruled out. Although it is perfectly reasonable to consider General Relativity to be an emergent theory, the theory it emerges from is (a) unknown (b) unobvious and (c) extremely difficult to take guesses at. Indeed, your offer of 1310.4691 is wholly rooted in this: canonically quantized GR conflicts violently with observations and experiments, and the usual workaround is to do some condition-fixing (which your referenced paper does) and then to try to get around the pseudo-forces brought in to describe local physics (in models like Page-Wooter these pseudo-forces appear as constraints in the theory ([2], which your authors reference in their first sentence and several times thereafter). The paper you point to also notes that the proposed experiment cannot select among a number of theories including General Relativity (where the Hamiltonian itself is a constraint).

> To me, time not being local is the controversial position.

I dunno, we do appear to live in an observable universe which admits an obvious equatorial 3+1 slicing in which there are an awful lot of Eulerian and nearly-Eulerian observers. Is the hill to die on the alignment of one's "natural" choice of timelike axis with the metric expansion or the way you put down coordinates on that axis? And how do you square either of those choices with the initial value formalism?

- --

[1] This is implicitly fixing a gauge wherein the surface of the moon is special; this is analogous to having a set of tunable air-pressure gauges at a point at sea level and setting it to 0 there, then using the readings of the tuned pressure gauges in helicopters riding above one another over the 0 point in order to say things about the state of each helicopter. In particular, one would use the reading of the pressure gauge as the basis of a coordinate axis (e.g. in marking coordinates on the radial axis in spherical coordinates on the 0-calibration point, or on the z axis in a choice of Cartesian coordinates on the 0-calibration point).

[2] K. Kuchař, in G. Kunstatter, D. Vincent, and J. Williams (eds), Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics, (Singapore, World Scientific, 1992).