Could the full theory of quantum gravity just be a nonrenormalizable quantum field theory?

You suggest that we can use a nonrenormalizible theory (NR) at energies greater than the cutoff, by meausuring sufficiently many coefficients at any energy.

However, a general expansion of an amplitude for a NR that breaks down at a scale $M$ reads $$ A(E) = A^0(E) \sum c_n \left (\frac{E}{M}\right)^n $$ I assumed that the amplitude was characterized by a single energy scale $E $. Thus at any energy $E\ge M$, we cannot calculate amplitudes from a finite subset of the unknown coefficients.

On the other hand, we could have an infinite stack of (NR) effective theories (EFTs). The new fields introduced in each EFT could successively raise the cutoff. In practice, however, this is nothing other than discovering new physics at higher energies and describing it with QFT. That's what we've been doing at colliders for decades.

The problem with GR+QM is that the counterterms include higher derivatives terms, $$ \mathcal L_\mathrm{ctr}\sim \partial^4h $$ where $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$.

Therefore, on accounts of the Ostrogradsky instability theorem, the system is unstable. This means that the whole program of perturbation theory makes little sense, and there is no reason for us to expect that the perturbative expansion has anything to do with what the theory is really telling us.

Therefore, QGR's perturbative expansion need not reflect what the non-perturbative theory is about. We just don't know what to do with the theory, we only know that perturbation theory cannot work.

For more information on Ostrogradsky's theorem and its relation to a possible theory of quantum gravity, see the very accessible review Avoiding Dark Energy with 1/R Modifications of Gravity, by R. P. Woodard.

To close this answer I would like to include a transcript from Strominger's essay Is there a Quantum Theory of Gravity:

The problem of unifying quantum field theory and general relativity was formally solved in 1967 by DeWitt in a series of classical papers [...]. Using standard principles of quantum mechanics, he constructed a unitary set of Feynman rules describing the quantum dynamics of the gravitational field. Unfortunately, the perturbation series in Newton's constant is not renormalizable. This is evident from power counting: the naive degree of divergence of an $L$ loop diagram is $2(L+1)$. Barring miracles, we therefore expect on dimensional grounds that a counterterm constructed from $L$ powers of the Riemann tensor will be necessary to renormalize the $(L+1)$th order of perturbation theory. Since a new coupling constant is introduced at each order, the theory loses its predictive power. This difficulty arises from the fact that the coupling constant has dimensions of inverse mass.

A first response to this problem was to conjecture that the full theory is in fact finite [...]. After all, the perturbation expansion is really an expansion in the dimensionless parameter $\kappa^2E$ ($E$ is an energy scale, $\kappa^2=32\pi G$). This expansion parameter is large at large energies. One cannot expect such an expansion to provide a systematic method for computing the effects of virtual gravitons of arbitrarily large energies. The non-renormalizability of the weak coupling expansion may just be due to a bad expansion of a good theory.

The finiteness conjecture has both intuitive and calculational motivations. On the intuitive side, there is a vague notion that one should not be able to propagate at energies where wavelengths are less that the Schwarzschild radius. The Planck energy should thus provide a natural cutoff. This notion has also received some support from explicit calculations. Non-perturbative summations of ladder [...] and cocoon graphs [...] have in fact produced finite results.

Unfortunately, these calculations are not gauge invariant and do not represent a systematic expansion in some small parameter. Furthermore, in recent years, systematic expansions have been developed in parameters that are small at all energies [...]. Quantum gravity is not finite order by order in these expansions.

The prospects for a sensible quantization of the Einstein action thus do not appear terribly bright. The next logical step is to alter the action by adding higher derivative terms such as $R^2$ [...]. In any case, one is forced to add these terms for renormalization.

This step is taken rather hesitantly because actions with four time derivatives generally describe theories that appear pathological even at the classical level. In fact, it can be shown that classical higher derivative gravity theories either have tachyons or negative energies for small, long wavelength fluctuations [...]. Thus they have pathologies that are evident on macroscopic length scales. How, then, can the quantum version of these theories possibly be reasonable? The miracle is that some, if not all, of these instabilities can be systematically eliminated from the quantum theory. This will be discussed further in the next section.

I really encourage the interested reader to have a look at the rest of the essay. In a nutshell, one can in principle formulate consistent theories of quantum gravity, but at a very high price. In fact, most people haven't heard about these formulations, which is a strong hint that the community doesn't consider them to be relevant nor useful solutions to the GR+QM problem. For one thing, they generate more questions than actual answers. But'll I leave the reader reflect about the situation themselves.

I think the reason people don't like this idea is that to keep the same physics at lower energy scales the coefficients of the nonrenormalizable terms should grow as you define the theory at higher and higher energy scales $\Lambda$. This is the flip side of being an 'irrelevant' interaction. They might become infinite at a finite value of $\Lambda$, in other words a Landau pole. In that case you could not define the theory at higher energy scales while keeping the same physics you already have at lower energy scales.

If they do not become infinite but in fact converge to a finite value, an ultraviolet fixed point, then the theory really is well defined. This is actually a serious proposal for quantum gravity called asymptotic safety.

Basically you can keep pushing the effective theory and either you start to see new physics, or it works all the way and you have asymptotic safety. The option where it really does break down at some finite $\Lambda$ would lead to people studying the regulator. Is it a lattice theory or what? That would really be new physics too.