I'm a freshman on Physics course, espite of this fact I have a quite interest on Gravitation. My question is:
QFT in curved spacetime is not viewed as a full quantum theory of gravity but rather as a first order approximation to it. It is defined only on background dependent scenarios, i.e. where the background metric is fixed. That means assuming that the quantum fields don't change the background metric, which shouldn't be the case in a theory of gravity.
Also in examples like calculation of black hole entropy, a quantum field theoretic treatment yields an infinite entropy (as opposed to the finite Bekenstein-Hawking entropy which is obtained semiclassically, for the full field theoretic treatment a distance cutoff is needed as in the brickwall approach of t'Hooft, which violates the principle of relativity), an issue which can be traced back to non-renormalizability of gravity as a field theory. The issue of non-renormalizability of gravity as a field theory also needs to be taken care of in a full theory of quantum gravity, as infinities are there due to loops in Feynman diagrammatic expansion which cannot be renormalized, and one requires an increasing number of counterterms at each loop so as to absorb up divergences. QFT in curved spacetime therefore isn't designed to be a quantum theory of gravity, but serves as a good tool and reveals some nice features using standard methods of QFT. It is viewed more as an effective field theory than a full theory.
In order for a theory of quantum gravity to be "alright", one looks at consistency checks which a quantum theory of gravity needs to satisfy. A theory of quantum gravity should yield calculations consistent with corrections to the Bekenstein-Hawking entropy.
There are a handful of approaches to quantum gravity as are given on this Wiki page. String theory and loop quantum gravity are the leading approaches, with other approaches too being practiced. Personally I believe string theory is a very strong candidate for a quantum theory of gravity, for it has been able to correctly satisfy various consistency checks (like the calculation of black hole entropy) with great success.