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tl;dr: coincident-events first, then labels (coordinates). [Einstein 1916, p.117 [2] although I remembered to look there only after writing all of the below]. One-way speed of light arguments are in danger of being coordinates-first, and thus insufficiently general for physics.[3]

The key word in your comment is

> directly

But why do we care? We have an abundance of indirect evidence, premised on direct tests of coordinate-independent features of our best most-fundamental theory. The two important features of (general) relativity are pointwise local Lorentz covariance -- where c is the only free parameter of the Lorentz group -- and the minimal coupling. Special relativity's Minkowski space is in this view a special static time-orientable spacetime in which we have global Poincaré invariance (c again is the only free parameter of the Poincaré group; the Lorentz group is a subgroup of the Poincaré group -- the latter includes all the spacetime translations, and in the Minkowski case the space-translation and time-translation symmetries all commute). When we go blithely parallel-transporting null vectors, this is what matters.

We can certainly write down an f(c) theory. Dicke did this in in his superb 1957 "Gravitation without a Principle of Equivalence" >https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.29....? and so have several others (See Ellis or Magueijo for a review <https://link.springer.com/article/10.1007/s10714-007-0396-4> corresponding with <https://arxiv.org/abs/astro-ph/0703751> resp. <https://iopscience.iop.org/article/10.1088/0034-4885/66/11/R...> open access, but corresponds with <https://arxiv.org/abs/astro-ph/0305457>).

It is far from silly to write down a theory where c varies in spacetime. It is the foundation of several alternative-to-cosmic-inflation decaying-bimetric theories of the very early universe, where c eventually stabilizes to its value in our local spacetime having been a different (typically much much much -- ~30 orders of magnitude -- higher) value during the formation of primordial matter density variations. The faster speed of light allows for distant reaches of the early universe to reach the same temperature with uniformity up to the small fluctuations in the cosmic microwave background.

Of course we run into the same point you've been working in this thread: it's hard to discover the exact function on c in the early universe. We have to rely on indirect evidence, and strong gravitational lensing is useful there. SVOM <https://svom.cnes.fr/en/SVOM/GP_mission.htm> is looking for Lorentz-invariance-violation (LIV)-induced modifications to the photon dispersion relation in vacuum, and is a particularly good platform for test of a Taylor-series expansion like E^2=p^2 c^2 ( 1 +- \sum_{n=1}^{\infty} a_n ), since GRBs at least somewhat escape the problem that the lowest order terms dominate at small energies, and they are distributed across the sky and at different redshifts. We are also now better equipped to study light echos (oh for a galactic supernova!) and detailed strong galactic lensing studies. Spoiler: the constraints on a spacetime-translational variation of c grow tighter with every observation. However, to fully rule out a sharp phase-change in c, we will need practical ~ 10^-15 Hz gravitational-wave astronomy. LIGO is most sensitive around 10^2 Hz; eLISA would be around 10^-2 Hz. (I am fairly sure the authors of most modern variable-speed-of-light early cosmologies knew as they were writing that they probably could hide in that hard-to-explore space through a few generations of gravitational wave observatories. One might say the same about a wide variety of recent cosmic inflation theories, too.)

In a general dynamical spacetime the notion of a two-way path is tricky. Even in Minkowski space, for a two-way signal, the return detection arrives at a different, later, point in spacetime than the outbound signal, even if the spacelike coordinates are always (0,0,0) [this is somewhat reminiscent of the twin paradox]. Outbound-and-return are two future-directed null geodesics. Your argument in this setting is equivalent to saying that we are somehow in trouble because the "outbound" and "return" null geodesics may have, without rescaling, different affinely-parametrized lengths.

In SR what we care about is that the signal is Lorentz invariant at each spacetime point where it could be sampled, even as sender and receiver/reflector are moving ultrarelativistically or are ultraboosted. Given Lorentz-invariance we can determine the three relevant points on the manifold. (Poincaré invariance means we can do this same test at any time or place in the flat space universe). Your complaint is that this is not a direct one-way measurement. OK, it's not. So what? We can in principle directly test Lorentz-invariance at any point (e.g. we can have a sparse gas with a well-understood (as in at the Standard Model of Particle Physics level) low extinction coefficient). If we have no flat space violation of Lorentz-invariance, we must have symmetry of light travel time for constant light-like separation.

In a dynamical general spacetime (Lorentz invariance -> local Lorentz invariance (LLI)), we can readily move the intended recipient of a one-way light pulse outside the reach of the light pulse itself; nature already does this for us in at least a couple of ways (metric expansion and astrophysical black holes). We can also have different delays on each arm of a two-way measurement, e.g. through Shapiro delay, around a spinning mass, or in the presence of a gravitational wave. However, at each point in the (vacuum part of the) curved spacetime [a] light obeys the massless wave equation and [b] local Lorentz invariance demands that the fraction of the wave at X propagates to a neighbouring point X' at c, and that X' must be drawn only from certain available neighbouring points.

So, really, it's not so much "what is the one-way velocity of light?" but rather "how much spacetime does a pulse of light traverse between two spacetime points?". Or in other words, we are looking for an affine parametrization on a curve of zero interval and that extremizes the length between two points on the manifold m and is constructed by parallel-propagating a tangent vector on m and in its own direction.

Consequently, I think the issue at the core of your points about measuring one-way speed of light is how to best label, with coordinates, two particular coincidence-points in a Lorentzian spacetime, rather than choosing two labels[1] (via your favourite synchronization scheme, for example) which are then used as the basis for a parametrization of a null curve. Then the question you ask is: "how can we know that the spacetime is Lorentzian?" or alternatively, "how do we know there is not some vicious additional gravitational field and vacuum polarization which makes the spacetime only seem Lorentzian?" for which experimentalists have generated a century worth of answers. The answer of how to best label two points in a Lorentzian manifold is \mu : it really depends on what and how you want to calculate. The physics here are that the two points (in your null-curve-parametrizing / one-way-light-travel-time experiment) are timelike-separated.