You asked a good question. Forgive me if my answer is aimed a little bit at the level of some of the other commenters in this thread (and discussion topic). I can do an ELI if someone wants.

General relativity simply equates the local density of the stress-energy tensor with the Einstein tensor (and possible scalar multipliers like the cosmological constant) at every point in the whole spacetime. If we measure curvature with e.g. Einstein lensing, we know what the stress-energy tensor must be. The stress-energy tensor encodes all moving matter, including internal non-gravitational degrees of freedom (DOFs). The ~seventeen fields of the Standard Model of particle physics fill all spacetime too, and contribute to the stress-energy tensor. Notably, invisible (in the sense that they do not couple to electromagnetism) DOFs exist in those fields and how they couple to each other; these standard-model mechanisms happily generate non-negligible stress-energy but nothing that we can see at a fine-grained level in any of our telescopes. Gluons' complicated self-interactions are the paradigmatic example: their whizzing about is most of the proton invariant mass, and so most of the mass of visible matter, and so most of the stress-energy in stars (gluon-gluon interactions is especially important in neutron stars, and neutron stars are an important diagnostic of any theory of gravitation) and molecular gas and dust; but gluons themselves are invisible and massless.

Dark matter, in a nutshell, says that "empty space" (ignoring the thin, cold relic fields of the cosmological microwave background and its neutrino equivalent) has some unknown internal degrees of freedom whose action generates stress-energy. Very broadly we call the generator(s) of that stress-energy dark matter.

The distribution of luminous matter (and different species thereof, and how it interacts (e.g. pressure, like ram pressure, can be important in galactic dynamics)) within a galaxy or cluster varies from galaxy to galaxy. Why shouldn't the distribution of dark matter?

So,

> possibly there is a deeper theory of gravity that explains both mercury's precession and galactic rotation

it's just General Relativity. The difference is that we know the distribution of stress energy within our solar system much better than we know the distribution of stress energy in much more distant, or much more complicated, systems (like galaxies or clusters of thousands of galaxies).

Alternatives to General Relativity broadly can take the approach that "empty space" (again ignoring the cosmic microwave background) is just that: there are no hidden non-gravitational degrees of freedom to discover there. Instead, stress-energy of the Standard Model as it is understood today generates the measured curvature seen in lensing by galaxy clusters like "El Gordo" (ACT-CL J0102-4915). Typically this is done by a redefinition of curvature to include auxiliary gravitational fields beyond the Einstein tensor, thus we get families of theories like tensor-scalar gravity, tensor-vector-scalar gravity, and so on. These fields couple with each other, so that the Newton-Einstein-like coupling in the bright parts of galaxies persists but the dynamics of that coupling generates auxiliary curvature in empty space outside the bright parts of the same galaxies. One often hears this described as introducing a new force or a "new fifth-force" produced by matter comparable to Newtonian's force of gravity; the new "force" has a different fall-off at a distance from the source matter than the 1/r^2 of Newtonian gravity or electromagnetism, and that fall-off is designed to produce Milgrom's low-acceleration relation. However (and in the spirit of a thread that started off with the words "generally covariant", MOND-compatible relativistic gravitation is still just gravitational fields being sourced by matter, but with more gravitational fields and interactions and self-interactions among and within those gravitational fields (with the weakness of gravitation proposed as cutting off these extra interactions' inducement of novel behaviours of bound systems like protons). Relativistic MOND approaches tend to be quite complicated in the gravitational sector, and of course must be no less complicated than general relativity (but for "dark matter") in the stress-energy sector.

In short, the GR approach is that the apparently empty space in front of a clearly distorted smaller-angular-diameter/higher-redshift/lower-brightness galaxy (like "La Flaca" <https://webbtelescope.org/contents/media/images/2023/119/01G...>) is not really empty -- there is stress-energy there that does not interact electromagnetically (it also sticks around unlike known standard-model particles that do not interact electromagnetically: photons, relativistic neutrinos -- those fly off to infinity rather than hang around in a halo structure). Studying these systems means understanding how the non-electromagnetic stress-energy is generated, and there are many hypotheses.

Conversely, the MONDian approach is that the apparently empty space around galaxies that flatten their rotation curve (and tbf there are MONDian approaches that capture some aspects of lensed objects) is really empty. Instead, the way that the standard model of particle physics generates curvature is different from General Relativity. Studying how gravity works for a particular (and more restrictive) distribution of matter is what relativistic MOND people set themselves up to do, and again there are also many hypotheses.

In both these broad approaches, the distribution of visible matter varies at different scales and in different systems at the same scale. Local matter distribution drives local gravitational phenomena. MONDian approaches roughly hold that there is no wiggle room -- gravitation must be generated by what we can see, and not by anything else. General Relativity is more lax -- there is no reason to expect that we can see every generator of stress-energy, or that the distribution of the generators must be uniform in space or in spacetime, and gravitational theory should not turn its back on the possibility of discovering new beyond-the-standard-model spacetime-filling matter fields.

One can also go really crazy and say that matter is not the (only or primary) generator of curvature, and that geometric curvature and matter simply coincide mostly by chance. A distribution of curvature that evolved in the early universe -- a cosmic gravitational background -- is taken to be relevant to the dynamics of galaxies and clusters about as old as ours. In these theories there are also auxiliary gravitational fields, and those typically allow for a MOND but are also designed to allow for the acceleration of the expansion of the universe and other features that relativistic MOND essentially imports from general relativity (in the form of a scalar multiplier on one or more of the curvature field(s)).

So, in summary I'll return to your first question:

> how [can] dark matter theories reproduce mondian dynamics naturally without each case requiring special tuning

They can't. There is an initial (or at least early) values surface that the laws of the theory turn into galaxies and clusters of all sorts. The initial values set up some distribution of interacting and self-interacting matter (that is initially, or evolves into, the standard model and dark matter). It also may set up primordial gravitational radiation not produced by the matter, and things like an evolving cosmological (non-)constant.

Relativistic MOND is not different on this front -- there must be some initial or early values surface that the laws of the theory turn into the standard model and the various gravitational fields around MONDian galaxies, mondian wide binaries, MONDian relativistic stars (like white dwarfs and neutron stars, especially in wide binaries of those), and so forth. These approaches may be even more sensitive to the distribution of things like primordial gravitational radiation because in general gravitational radiation is considerably more complicated by the presence of fields beyond the Einstein tensor: any additional scalar, tensor, or vector field may have their own polarizations, and their polarizations may modify the polarization of the mathematical structure closest to that of the metric tensor or the Weyl tensor parts of the Einstein tensor. Thus the considerable interest in the small variations in the cosmic microwave background by people who spend any time comparing General Relativity and relativistic MONDian theories.

Ultimately the "special tuning" is the "past hypothesis", which is highly similar for both approaches: the distant past of the universe was hotter, denser, and lower entropy than the universe full of galaxies. The lower entropy is the puzzle. What ordered that?
Again, there are many and various thoughts, and a handful of plausible hypotheses.