> Maybe for some other reason

The usual objection is that test stress-energy fairly generically forms singularities (via Raychaudhuri focusing and Jeans instability mechanisms, for example). Singularities obstruct the determination of the entire spacetime from a "complete sample" of it across a spacelike hypersurface, which makes initial values surfaces approaches containing singularities (very probably) incomplete. The genericity means that most spacetimes are incomplete, and if one thinks a fundamental theory should be able to describe a test universe completely (i.e., every field value everywhere and at every time), this is a problem.

This theoretical problem appears already in a purely classical stress-energy on a Lorentzian spacetime. For all practical purposes this only matters for carefully defined test spacetimes that we care to simulate, rather than for real astrophysical systems in our (Lorentzian) universe.
After all we can only describe mathematically-isolated parts of our universe, and with current technology we can only reasonably describe even those with approximations and effective theories. There is no shame in using Newtonian gravitation when planning a moon shot even when you know that you can't use Newtonian gravitation to describe the entire universe or even parts of it like the precession of Mercury's orbit much less the Hulse-Taylor system. There is no shame in using General Relativity in satellite-based navigation systems, either, even if you suspect GR cannot describe some feature of our universe much further afield.

So, let's consider a concrete theoretical example using a Schwarzschild black hole for which we break the time symmetry (i.e., we let its mass vary over time) and whose mass we drive with some classical matter that only negligibly perturbs the exterior part of Schwarzschild spacetime otherwise. With the gentle assumption of the no-hair conjecture being true, one can toss a spherically symmetrical shell of perfectly classical matter (remember this is a probe of theory, rather than a simulation of reality) of mass M into a black hole at the centre of the shell, or two concentric shells of classical matter of mass M/2 each, or three concentric shells of classical matter of mass M/3 each. If one takes a initial values surface (IVS) in the future of this shell-tossing and works backwards to determine the predecessors value surfaces of the IVS, one cannot decide whether one, two, or three shells were thrown in. The black hole singularity at r = r_0 is said to have destroyed the information, or if you like, the information is simply not encoded in the modified Schwarzschild solution described above.

That this is a problem pretty generically in nonvacuum Lorentzian spacetimes is sufficient to make General Relativity, at least in its initial values formulation, a less-than-desirable candidate for a fundamental theory of our universe from which other theories (Newton, Einstein-Maxwell, QCD, Navier-Stokes, etc.) might be derived.

Indeed, taking the position that one might erase quantum weirdness in strong gravity (and not worry much about exactly how by resorting to the slightly-less-gentle assumption of a cosmic censorship conjecture being true, such that strong gravity is always completely enclosed by a horizon) does not repair this problem, even if it might conceivably fix all the questions raised by Hawking radiation. Essentially, you're still left with questions like (simplifying the nuclear physics): given an initial values surface for black hole M at time T_now, when considering time T_slightly_earlier for black hole M_slightly_lower, can one work out whether the increase in mass was due to two deuterons falling in or one helium nucleus falling in? That is, we just have a quantum version of the classical shells above, and the same failure to extend from arbitrary IVSes across a whole fairly-generic spacetime.

We can make it worse by taking a surface T_far_future and M_much_smaller, that is after Hawking radiation has much reduced the mass of our black hole, and discover that we have an apparent violation of unitarity: our quarks and gluons (and electrons and so on) can't still be in the black hole (it's now too light to hide them behind the horizon) and yet there is now a cold gas of photons "carrying" the difference in mass. Lots of very clever people continue to study this "AMPS firewall" problem, which is essentially a theoretical sharpening of the "we lose the ability to describe the entire toy universe when a black hole is involved, and the real universe appears to have black holes in it" problem.

Experiments which provide evidence favouring a generic blocking mechanism that prevents matter from forming gravitational singularities in General Relativity would undoubtedly encourage a great deal of research into GR as a candidate fundamental theory once again. (Geometry-is-fundamental-as-in-stress-energy-is-geometrical-when-you-look-closely extensions of GR are still not dead-ended; just unpopular compared to stress-energy-is-a-specific-gauge-theory-and-geometry-probably-arises-from-it approaches).

Penultimately, a mathematical discovery that proves that gravitational singularities in GR are procedural artifacts arising from things like gauge fixing, and that they vanish with a good choice of gauge (or coordinate condition or whatnot) much like the coordinate singularity at r = r_s in the Schwarzschild solution was eventually found to vanish under different choices of coordinates (none of which makes the infinite Kretschmann scalar (at r = r_0 in Schwarzschild) go away).

Finally, we can just shrug and keep working with GR (and careful approximations) where it's useful and obviously in concordance with observations, and not sweat the fine details we can't in practice measure now anyway. If someone discovers an observational conflict which undermines GR as a theory in specific circumstances, it seems unlikely that GR will stop being used or taught any more than Newton has been abolished from classrooms and lecture halls, mostly because it is such a brilliantly effective theory for why there are very specific and observed-in-detail deviations from Newtonian gravitation. Practically nothing comes anywhere close to the match between GR and experiment and observation, and of those that do, most do worse than plain old Newton (sign reversals are rife) in certain situations : https://en.wikipedia.org/wiki/Alternatives_to_general_relati...