To start with, there is no canonical approach to quantum gravity, rather, there are different quantisation procedures (as for all the other interactions), which do or do not fail for this or that other reason without having to go to black hole entropy.
The standard quantisation procedure by means of the path integral fails simply because it is not renormalisable: everything works (somehow) fine but the coupling constant does not have the right dimensions required for a theory to be renormalisable perturbatively. There is no critique, it is just what it is and you can find reasonable walkthroughs in any textbook on QFT (Ryder, for instance).
As a workaround, since the path integral terms do not seem to be polite enough, some additional models have been proposed. Loop quantum gravity and spin foam models (many papers by Oriti, see for example an introduction here, or by Perez, see here and references therein) happen to be very promising, because they reduce the complexity of the infinite degrees of freedom discretising the topology of the space time through triangulations: states and functionals are then defined on the knots networks and scattering amplitudes can be computed in terms of irreducible representations of some Lie groups. Unfortunately, in four dimensions such representations happen to be infinite dimensional and thus problems of renormalisability still occur. A branch named Group Field Theory is being developed only using Lie groups representations to write down all the observable quantities of interest (for instance Krajewski) but still at some point they fail upon some infinities (mostly due to Lie groups being non-compact).
Other theories try to only quantise some linear terms on the right hand side of the Einstein's equations, but besides being very particular effective theories, they too have problems of renormalisability in correspondence of some divergences.
Quantisation procedure via Fock space would not work, being the equation of motion non-linear.