Ok, I think there are a couple ways of digging out a question to answer from your comment as I understand it.
I believe you are asking about how to solve the trajectories of electromagnetic radiation generated just outside these central black holes, since essentially that's what determines the images released to the public today.
I'm going to restrict this to the "lens" of the production of hard X-Rays and gammas around a black hole by inverse Compton scattering. https://svs.gsfc.nasa.gov/vis/a010000/a011200/a011206/index....> has a pretty couple of visualizations. (It is not a coincidence that the swirls vaguely resemble some of the images that were revealed in today's ESO presentations.)
Pretty much nobody is using exact analytical solutions to the Einstein Field Equations of General Relativity to predict central black hole observables. Instead one uses a combination of numerical methods https://en.wikipedia.org/wiki/Numerical_relativity> and approximations to the full Einstein Field Equations including linearized gravity, the effective one body formalism, and post-Newtonian expansions (the wikipedia article for which has a handy chart of the domain of applicability for these https://en.wikipedia.org/wiki/Post-Newtonian_expansion>). One can do standard model physics set against any of these formalisms (or against several as things plunge inwards and/or climb outwards from the near-horizon) and get useful results.
If one sat down (as a theorist) and were to grind out an exact analytical solution (this would have to be for a very tiny sample of light-producing events to be tractable cf. [1]), one would find there is no need to make quantum corrections to the gravitational side of the Einstein Field Equations. The reason for this is that General Relativity guarantees a small patch of flat spacetime around every point everywhere. As long as the "small patch" is big enough to enclose an electron-gamma scattering event, there is no need for quantum corrections. This translates in practice to not having to introduce higher-order terms "correcting" the formalisms above for strong gravity, and in fact partially justifies each of those.
Where we worry theoretically is when spacetime curvature nearby is so strong that the "small patch" starts being smaller than a gamma ray. Smaller can be read as a combination of spatial extent vs wavelength or longer than the half-period of the frequency. When that happens, we have to mathematically stabilize the spacetime around the electron-gamma interaction in order to use the Standard Model's description of the scattering, and then we have to figure out how to undo the stabilization so the emitted photon has the right energy.
We would want to do this by adding in quantum corrections to whatever gravitational formalism we are using. These are easiest to see as additional higher-order terms added on to the Taylor-series-like post-Newtonian expansion.
It turns out that the strength of the local spacetime curvature (and thus the inverse of the extent of the "small patch" of flat spacetime: stronger curvature, smaller patch of flat space) outside even stellar-mass black holes is much larger than we need for pretty much any Standard Model physics to be feasible without -- or with only very gentle -- quantum corrections. For supermassive black holes, local spacetime curvature just outside the horizon is smaller than for stellar black holes, so the local patch of flat space everywhere near the black hole is much larger than that in any particle physics laboratory here on Earth. Since the tidal effects of Earth and the sun don't make much difference to physical experiments done at e.g. CERN, the even gentler tidal effects of Sgr A* and the weaker still tidal effects around M87* can basically be ignored.
Where do we start needing significant corrections, and start having to think about not using some of these formalisms instead of harder and harder work designing numerical methods based on the full theory of General Relativity? (For example, we might end up having to add many many many higher-order terms to our Taylor-series-like post-Newtonian expansion, each adjusting by something like a tiiiiiny 1/c^{ever larger number}). The answer: it's tractable until we are deeeeeep inside the event horizon, where we can't see the results of what's going on from outside. Very near the singularity the expansion approach starts requiring millions, billions, billions-to-the-power-of-billions of additional small correcting terms to retain accuracy, and it's a losing battle, even with mathematical tricks to shrink the number of terms and/or sizes of exponents ("renormalization", which is out of scope for this answer). At the singularity, this approach can only fail. Far from the singularity, but within the horizon of a large black hole, it works just fine. And in any event we only really care about what's outside the horizon, because we can't interact with anything inside: it just leaves no imprint for our telescopes to detect.
General Relativity and its approximations work perfectly well outside Sgr A* for known particle physics (and even some higher-energy extensions to the Standard Model).
Today's results fail to support several alternatives to General Relativity that correspond to a need for quantum gravity corrections just outside the horizon of Sgr A*. Among them are theories which predict "bouncing" or "reflecting" surfaces, and radiating compact stars (e.g. quark stars, boson stars -- things that are even more compact than neutron stars, but held up from collapse by an as yet undiscovered degeneracy pressure as in https://www.einstein-online.info/en/explandict/degeneracy-pr...> for electrons).
So, in other words, there is no need for a theory of quantum gravity for the findings made public today. (The findings do cause possibly fatal trouble for alternative theories of gravity that expect quantum effects just at the horizon of Sgr A*.)
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[1] a discussion of how this works, and a neat simulator, for one photon around a Kerr black hole: https://duetosymmetry.com/tool/kerr-circular-photon-orbits/>