It depends on how you define "local observable" and that - in turn - puts the focus of the question on which definition is physically relevant! The matter is discussed more generally in Local and gauge invariant observables in gravity, Khavkine.
I'm not keen on folklore results, which this has practically descended into, nor on hand-waving arguments, and the author strikes the wrong chord at the outset with "It is well known that" - which is the standard herald for folklore and (which I say as a reviewer:) is reason to return the paper for modification. However, the author did actually defer more in-depth discussion of the topic to their references 16 and 7 (respectively, Giddings', Marolf's and Hartle's Observables in effective gravity and Brunetti's, Fredenhagen's and Rejzner's Quantum gravity from the point of view of locally covariant quantum field theory).
The latter authors, particularly Brunetti and Fredenhagen, are well-known for having jumped on the causal perturbation theory bandwagon and taking it a step further with their microlocal analysis framework, and then attacking the problem of quantum gravity with this in a way that by-passes the issue of non-renormalizability. This looks like it's the latest in a series, but I haven't been paying attention on their progress since early this century. I might have to look at this more closely.
But, anyway, the key issue is: how do you actually define local observables? Khavkine is directly addressing the issue of physically relevant definitions, providing a better formulation that gets the job done. To quote the concluding part of their abstract:
Moreover, generalized local gauge invariant observables are sufficient to separate diffeomorphism orbits on this admissible subset of the phase space. Connecting the construction with the notion of differential invariants, gives a general scheme for defining generalized local gauge invariant observables in arbitrary gauge theories, which happens to agree with well-known results for Maxwell and Yang-Mills theories.
The predecessors to their idea, as they point out, go back to the early 1960's: Bergman and Komar. Section 2 of Khavkine lays out the standard definition that leads to the no-go result. Section 3 provides the modification that overcomes it. Briefly scanning of section 2, I see that everything is being framed in terms of $C^\ast$ algebras, which transcends the distinction between classical and quantum, so it looks like it's something that cuts across the board and is not specifically quantum.
The paper looks like it's an interesting read, because Khavkine is more closely in line with the standard treatment of field operators, i.e. one where points $x$ are smeared out as compactly-supported functions $f(x)$, with $φ(x)$ replaced by $φ(f) = \int φ(x) f(x) dx$. They're called "test functions" (a standard canard in theories of generalized functions, like Schwartz or Columbeau) but are better thought of as "smeared points". The discussion in section 1 will also help make Brunetti and Fredenhagen's work more accessible, since they use the kind of math in their work.
I'll probably have more to say on this later, either in comments or a later edit, after closely reviewing the paper and its references; since the whole issue of diffeomorphism symmetry and gravitation (particularly in the context of quantum theory) is something I actively sought out some information on, rather than something that came to me. It's a Pull Issue, with me, not a Push Issue.
In the context of quantum theory, one thing you should pay attention to is that any proposal for somehow "quantizing" geometry or even discretizing it, automatically breaks diffeomorphism symmetry, with loop quantum gravity being a borderline exception. In addition, all of the regularization approaches that I am aware of ... (discrete) causal sets, led by someone who I knew, who used to be at my alma mater; Regge calculus, which might be considered a predecessor of causal sets; Lattice Field Theory, which is used to regularize quantum field theory ... all break diffeomorphism symmetry. An example of something that might not (quite) break diffeomorphism symmetry would be a method of "third quantization" that quantizes the Second Noether Theorem (English version), instead of the geometry. That gets closer to the direction Brunetti and Fredenhagen were moving in, the last time I checked on their progress.