To call something "quantum gravity" at all, it needs to be a UV complete theory which has Einstein gravity as its effective theory at low energies. So for the questions "can quantum gravity always be approximated by an EFT" and "can quantum gravity always be equivalent to an EFT", the answers are "yes" and "no" respectively. Both by definition. Note that the basic consistency check of reducing to GR at low energies is already something that loop quantum gravity has trouble with.
Exact equivalence to a CFT is another story. There is currently much debate over flat and dS backgrounds. But for quantum gravity around a background that is asymptotically AdS, it is widely believed that some CFT dual is inevitable.
The modern approach to AdS/CFT which starts with QFT in AdS (example reference) is enough to show, just from geometry, that the boundary limits of local operators in the QFT will have conformally covariant correlators. It is then quite believable that making gravity dynamical in the bulk will give this system of correlators a stress tensor and turn it into a CFT. So we can say that being able to describe asymptotically $AdS_{d + 1}$ quantum gravity theories by a $CFT_d$ is another consistency check. String theory passes this check and moreover provides enough microscopic information to suggest which CFT you will get in many cases.