I am not sure, if this contribution can help you and adds new info here. It is an excerpt from notes that I took for myself. It is a prosaic collection of arguments, that I found. I wanted to motivate myself properly for the stuff, that I was studying at the end of my undergrad time, at the level of understanding that I had back then.
I read somewhere else, that you are sometimes irritated by the imprecision of the physicists. Mathematical rigour will never compensate for the lack of empirical basis. I would like to encourage you to absorb as much as you can from the meta-theoretical vantage point that philosophers of science especially physics takes. Certainly, they are not physicists themselves, but they add an important ingredient tot he soup. They are unbiased and challange the physicists assumptions, convictions, ideology and so on.
A careful analysis of current fundamental theoretical physics leads to the observation that we have two mutually incompatible theories, Quantum Theory and General Relativity and together with a third framework, Thermodynamics and Statistical Mechanics, one is rather successfully able to describe a vast domain of physical phenomena.
Despite this empirical effectiveness, we have to state that fundamental physics is in a conceptual confusion regarding the above mentioned theories’ elementary and universal assumptions.
On the one side stands a manifold of Quantum Theories and Quantum Field Theories, which are used to describe the phenomena of physics on a small scale.
For example, QFT is applied in the Standard Model of elementary particle physics, where it describes the fundamental interactions of the electromagnetic, weak and strong forces.
On the other side is General Relativity, which is used to depict the structure of the universe on a large scale. It describes the fundamental interaction between matter and spacetime geometry by means of Einstein's field equations
Both these theories are empirically well supported within their own domains of application and description of physical phenomena but the question arises if an overlapping domain exists and if so, how we could describe and observe it. At the so-called Planck scale it is expected that an interface of both theories is needed to provide a satisfying description of the microstructure of spacetime (together with matter). A theory, which would be able to do so, is called Quantum Gravity.
As of today, there is still no complete and consistent quantum theory of gravity available and all candidates suffer from formal and conceptual problems. Their biggest common problem is the difficulty to test their predictions by experiments, since the relevant scale is (probably) too far away for current experimentation.
This is certainly not a strong counterargument against research in this field, because future data from cosmological and astrophysical observations become sooner or later available and will provide the needed guidance to modify or dismiss the approaches, which we have or construct overall new ones.
Let us conduct an anamnesis of the problem of the mutual incompatibility of GR and QT.
Observe at first, that in QT one uses an external time parameter to describe the evolution of states through e.g. the Schr\"odinger equation. In QFT one assumes a fixed non-dynamical spacetime background as a part of its ontological basis, being independent of the fields, which are dwelling on it.
In GR one dispenses with the notions of external time and fixed spacetime background. Instead a dynamical spacetime geometry is introduced, which is identical with the gravitational field. The coordinate time becomes a non-observable gauge variable and the physical variable called 'time', is theorized by a function of the gravitational field.
There is an interplay between this dynamical background structure and the matter fields. This leads to their interwoven co-evolution, captured by the field equations of Einstein
There one observes that the left hand side deals with the spacetime geometry, which has a smooth classical structure and on the other side the energy-momentum tensor of matter fields is given. These fields are fundamentally quantum mechanical.
This stipulates that at small scales all dynamical fields and physical quantities should have quantum properties. This means that the fields are composed by discrete quanta, they follow the superposition principle and that they obey probabilistic laws. Physical quantities don't have sharply defined values in general, which is captured by Heisenberg’s uncertainty principle.
GR with its smooth Riemannian metric field and deterministic form violates this. Physical quantities are depicted by tensor fields, which usually have well-defined values. The energy-momentum tensor should be an operator, revealing an inconsistency of the formalism. One would have to replace the operator by an expectation value, depending on a fixed spacetime background, though the idea of the field equations is to have the metric dynamical.
In addition, gravity couples universally to all forms of energy. Since this energy is quantized the coupling should be so, too.
A quantum theory of gravity should therefore be capable of synthesizing both incommensurable frameworks and resolve these conceptual problems by giving a correct description of quantum geometry and matter.
Penrose and Hawking have proven that there are inevitable spacetime singularities under reasonable conditions on causality and energy, which cannot be prevented. This is articulated in the singularity theorems, referring to the assumed initial singularity of the cosmos and black holes. The consequence is, that General Relativity cannot be a valid theory without restrictions and one expects that in domains of strong gravitational fields as singularities, a theory of quantum gravity should replace General Relativity and explain e.g. the evolution of black holes. Hence, there seem to be a class of phenomena, which should be explained by means of generally relativistic and quantum effects.
Black holes radiate with Hawking-radiation, which is a hypothetical result obtained from QFT on curved spacetime. This is a semi-classical theory, which claims that a classical and quantum sector can coexist, with which one means, that the matter fields are quantum and the gravitational field stays classical. Its descriptive power breaks down e.g. at the final stage of black hole evaporation. In addition, it is well-known that QFTs suffer from UV divergences and one expects that a theory of quantum gravity, should provide a cut-off scale and thus cure these divergences.
The natural scale, where effects of quantum gravity are expected to occur, is the Planck scale, expressed by units given by Planck in 1899. The size of the Planck scale, if it is not an extrapolation, makes it extremely difficult to design experiments, which could test the few predictions of the different proponent theories of quantum gravity.
An ontological synthesis of the principles of GR and QT could lead to a general covariant quantum field theory, assuming that the ontological principles of these theories hold up to the Planck scale.
The idea, that there might be two separate phenomenological domains for GR and QT, respectively, with an empty intersection and that there is then no need for a theory of quantum gravity is however plagued by the fact that the interaction between a classical and quantum system is inconsistent.
There are also meta-theoretical arguments for establishing such a theory. For example one could expect, that a theory that unifies the concepts of GR and QT should be stronger in explaining established facts in addition to make own new predictions.
Often the argument of unification is rolled out, which is connected to the reduction of complexity and aims at finding a single coherent framework. Aiming at unity could e.g. mean the unity of nature, i.e. that the nature has a unified structure and one expects that it allows for the systematic description of the parts, which are empirically accessible, scientific method, i.e. that there is a unique way to generate scientific knowledge and/or of theory, i.e. that scientific theories should be unified concerning the theories' terminology, ontology and nomology. These arguments are certainly sufficient for the motivation to search for a theory of quantum gravity but they are certainly not enough to imply that it is necessary to quantize gravity in order to achieve this aim.
However, the necessary arguments for the quantization of gravity are provided by physical theories, which was elaborated above.
One could also take the different standpoint and argue that the gravitational field does not need to get quantized. Proponents of this perspective argue that gravity should be seen as some sort of induced/emergent force, not being fundamental. One idea could be, that gravity emerges from thermodynamical considerations, being a collective phenomenon. Considering such a stance is certainly justified, if one argues that gravity seems to be a profoundly different interaction in comparison to the other three fundamental forces, described in the Standard Model of particle physics.
However, ideas like this disregard e.g. that it is possible to quantize also collective degrees of freedom (like phonons) or they are ambiguous about seeing the metric field as being dynamical.
For long time there has been ongoing research on the topic of quantum gravity resulting in various approaches, like String Theory, Loop Quantum Gravity, Causal Dynamical Triangulation, Non-commutative Geometry and Causal sets and others. This manifold of approaches can e.g. be classified by choosing the relative weight of QFT and GR as a classification criterion.
Initial attempts at quantizing gravity by means of techniques, which were developed for quantizing the other interactions, tried to covariantly quantize gravity, which needed the introduction of a splitting of the gravitational field into a fixed flat background metric and a massless perturbation, yielding a linearisation of Einstein's field equations and leading to gravitational waves as their solution. One set out to quantize these waves, calling the perturbations 'gravitons' and GR appears to look like a theory for a massless spin-$2$ field, propagating on flat Minkowski spacetime.
However, it was argued that covariant quantizations of GR are not perturbatively renormalizable, standing in contrast to Yang-Mills theories. One could argue then, that this approach did not work out, since one should have considered the full metric to be quantized and in addition that a granular structure of spacetime should circumvent the problem of renormalization.
Other ideas propose to see GR as an effective theory, not being the right one at smaller scales than the ones, where it finds its strong empirical support.
Another idea is to consider the canonical quantization of GR starting from a constrained Hamiltonian system. It was argued that bringing GR into Hamiltonian form would break the manifest general covariance/ diffeomorphism invariance due to the '$3+1$'-splitting of spacetime but others disagree about this criticism. This has led to quantum geometrodynamics and more recently to LQG. LQG attempts at constructing a solid mathematical, non-perturbative and background independent general covariant quantum theory of GR. There it is believed that the theory of quantum gravity should be a quantum theory of spacetime geometry incorporating diffeomorphism invariance. The idea is to write GR in the Hamiltonian formalism of a diff-invariant Yang-Mills field theory, with compact gauge group. One does not assume an a priori background metric. Problematic with this approach is mostly the poorly understood dynamics, which motivated the development of the so-called spinfoam models. The open question remains, whether it is possible to derive a smooth spacetime in an appropriate limit from them.
The failure of current physics to establish a unique and straightforward way to a quantum theory of gravity suggests that the formulation of such a theory requires new formalisms and physics.