I disagree: Relativistic frequency shifts (doppler shift, gravitational shift, cosmological shift,...) can all be understood kinematically:
The invariant quantity you have to consider is photon 4-momentum, with energies at time of emission and absorption given by the projections onto the 4-velocities of the respective observers, so it's all about the relative orientation of some vectors, ie kinematics.
If spacetime is flat, photon momentum is constant, so all that matters is the relationship between emitter and observer frame. However, in curved spacetimes, there's no distance parallelism, and photon momentum at time of emission and absorption will be related by parallel transport along the photon geodesic.
In principle, you could even view the process of parallel transport as a series of infinitesimal Doppler shifts, though I would argue that puts emphasis on the wrong thing (the arbitrary frames chosen along the geodesic).
I'm not sure if you are objecting to approaches like the tetrad formalism in your final sentence, or even to gauge freedom itself. There's no reason an observer -- even an always lightlike one -- can't carry around its own system of coordinates. People from ISS astronauts (LVLH vs ECEF/ITRF) to aerobatic pilots (LVLH/LENU to GPS) to pedestrians looking at map apps on their smart phones do that casually all the time, and nobody expects ensembles of objects to move with respect to each other as one swaps sets of coordinates. Setting a new coordinate system every few steps while navigating on foot through a complicated city core is perfectly fine, and so is setting up a new coordinate frame at every infinitesimal step along an affine-parametrized curve traced out by a photon. IMHO that is the core of relativity.
I'm not sure if you are objecting to approaches like the tetrad formalism in your final sentence
The final sentence was just a comment on the pedagogy of deriving cosmological redshift via infinitesimal Doppler shifts. There's nothing wrong with it (sections of the frame bundles and all that jazz if you want to get formal), but to me, this feels like putting things on their head: I consider the Doppler shift a special case of the generic relativistic shift, making it circular to then explain the generic case as a series of infinitesimal Doppler shifts integrated along a photon's worldline.
No, it's not. The (local) metrics sourced by the Earth, Earth-Moon, Sun, solar system, Milky Way, and the Local Group are not at all like FLRW with or without a cosmological constant, while all those metrics are very much like Kerr or LTB, wherein ordinary matter like gas and dust gradually condenses into a point. Conversely, the metric globally is very much like FLRW with a small positive cosmological constant, and very much unlike a spacetime in which matter condenses into a point.
We can use Israel junctions to knit together "stacks" of regions that are well-approximated by a Kerr or LTB metric, and ultimately knit galaxy cluster sized LTB regions into expanding FLRW, representing the LTB regions as dust particles. This matches observations very well, and is called a "swiss cheese" cosmology, the name arising from how gravitational collapse ultimately creates voids -- holes -- in the otherwise homogeneous diluting dusts.
Kolb and others have a decent overview of how an Einstein-de Sitter-Lemaître-Tolman-Bondi swiss cheese universe works (the paper is in the context of the homogeneous-vs-inhomogeneous+backreacting cosmology debate, which is largely about connecting theory with astrophysical observations of a somewhat lumpy real universe especially in the transition from the matter-dominated era to the dark-energy dominated era) : https://www.osti.gov/biblio/21023997-cosmological-observable... ( PDF https://archive-ouverte.unige.ch/files/downloads/0/0/0/3/6/5... ). There's a good textbook treatment in Harwit's _Astrophysical Concepts_ as well (a preview exists at https://books.google.co.uk/books?id=gZfuBwAAQBAJ&pg=PA516&lp... ), and MTW has a (not very easy) overview of Israel-Darmois junctions in §21.13.
Such swiss cheese models match observations so well that it would be surprising (in a cool way) to find that there is any metric expansion at all in our solar system or in any of the galaxies and galaxy clusters we observe on the sky. We don't see that, though. In particular, one should take into account the evidence for accelerated expansion over the past few decades. Given that, one can consider a "fifth-force" (or nth, given Higgs etc.) representable as a field with a density and gradient, rather than the purely geometrical scaling in \Lambda-CDM. (This is at the root of some quintessence projects.) Results usually involve a lot of dynamics to suppress cosmological shear and other "little rip" effects at the margins of clusters with highly-luminous components, and adding several degrees of freedom that lack other physical support seems a lot worse than accepting an Einstein-de Sitter like swiss cheese geometrical model.
> local reference frame
It has nothing to do with reference frames; the metric tensor focuses matter and light one way or another (cf. Raychaudhuri) and within galaxy clusters practically all matter is focused towards a point in the future, whereas globally the same matter, and all other matter, and light, focus towards a point in the past (and the future focus points within galaxy clusters do not themselves focus together in the future). The only requirement is that the manifold is Lorentzian and time-orientable; you can use any system of coordinates that an arbitrary observer carries around with it, and you only need to do that if you care to describe when in the past or future and/or where relative to an observer the focus points lie.
> ... to match a distant galaxy's ...
How do you propose to do that? I think it's worth it for you to think about that a little: does your conception of a frame in which \lambda_{obsv} = \lambda_{emit} (rather than 1 + z = \frac{\lambda_{obsv}}{\lambda_{emit}}) extend into the distant past and distant future? How accelerated is this z-suppressing frame? In this frame what's the expectation for the wavelength of a CMB photon over time, or for galactic and extragalctic H I lines ?
> doppler expansion due to the nature of differing relativistic reference frames
Kinematical interpretations fall apart really spectacularly at high z, and conveniently we have a probe of (6 > z > 1) in the https://en.wikipedia.org/wiki/Lyman-alpha_forest which is a challenge for constructing a sensible z-suppressing frame of reference.
(See also MTW §29.2)