The notion of a quantum field as being an operator valued distribution on a fixed space by design only applies to QFTs that are defined on a fixed such background, or at least on all possible such backgrounds, one at a time, as in AQFT on curved backgrounds.
But there are more general definitions of "quantum field theory" that do not rely on this assumption.
Notably there is the "FQFT" definition which asserts that a quantum field theory is a rule that assigns spaces of states to abstract (n-1)-dimensional manifolds (not equipped with metrics or anything) and assigns correlators to n-dimensional cobordisms between these. This is indeed the axiomatics which is understood to a very strong degree for the special case of QFT of which quantum gravity is supposed to be an example, namely topological quantum field theories, whoch observables etc. do not depend on any fixed background metric.
Since rather recently, cobordism theorem completely classifies the local quantum field theories of this type. but already since long ago some people are trying to guess/construct those [4d TQFTs](ncatlab.org/nlab/show/4d TQFT) which have a chance of being the end result of the quantization of Einstein-Hilbert gravity.
While there are various proposals, nothing definite is really known. But if you want to get a feeling for what a 4d topological quantum field theory such as 4d quantum gravity (if it exists) might look like as a formal object, it may help to look at these examples. The most widely followed approach goes like this:
It is pretty well understood how 3d TQFTs such as standard 3d Chern-Simons theory are induced from Hopf algebras, hence from certain bialgebras, and so one can try to mimic this in one dimension and one categorical degree higher by using trialgebras and Hopf monoidal categories.
While lots of things can be constructed this way, for none of them is it clear how to show that the 4d QFT defines this way actually "is the quantiatio of Einstein-gravity", and it is not clear if any such exists at all in this fashion.
On the other hand, to put this in perspective notice that also the statement that 3d quantized Chern-Simons theory -- about such a comparatively huge body of mathematically powerful literature exists -- is indeed given by the 3d TQFT functor constructed by the Reshetikhin-Turaev model or the Turaev-Viro model is widely believed and highly plausible, but not really fully proven, as far as I can see.