Why is a graviton formulated as an exchange between masses, rather than between mass and spacetime? In Sean Carroll's Lecture Notes on General Relativity he states:

The gravitational interaction, meanwhile, can be thought of as due to exchange of a virtual graviton (a quantized perturbation of the metric). The nonlinearity manifests itself as the fact that both electrons and gravitons (and anything else) can exchange virtual gravitons, and therefore exert a gravitational force:


Why would gravitons exchange information directly between particles? Doesn't general relativity show that gravity is a secondary result of mass affecting the curvature of spacetime, not a direct interaction between masses?
I.e, why is it (mass)->{graviton}->(mass), and not (mass)->{graviton}->[spacetime]?
[Note: This question is not whether gravity is a force or not, but why a quantization of gravity would be formulated as an exchange between particles instead of between masses and spacetime itself (as general relativity seems to indicate).]
 A: One could equally well ask "why is QED formulated as $\text{charge} \to \text{photon} \to \text{charge}$, and not $\text{charge} \to \text{photon} \to \text{EM field}$?"  The answer is that photons are the quantum-mechanical excitations of the EM field;  trying to draw a distinction between "photons" and "the EM field" is not terribly fruitful, since they are really two ways of looking at the same thing.
Similarly gravitons are the quantum-mechanical excitations of the metric on spacetime (or they would be if we had a self-consistent theory of quantum gravity.)  Trying to draw a distinction between "gravitons" and "the metric" is not really fruitful for the same reason.
A: The gravitons are excitations on spacetime itself. In this sense, matter particles exchanging gravitons means exactly they are moving closer together due to spacetime effects. Furthermore, notice that gravitons interact with other gravitons, and hence you can also have matter particles exchanging gravitons with other gravitons, which is an exchange between mass and spacetime.
By the way, do notice the graviton is not formulated like that. What you actually do is to pick a spacetime, add perturbations, compute how these perturbations behave using your favorite theory (typically General Relativity) and then quantize that. This interaction is derived from other principles, not proposed in an ad hoc manner.
A: Part of the Deus ex machina here that supposedly resolves the circular definition is that mass, for purposes of many theories of gravity, can be of various kinds.
Many theories argue that inertial mass for example is different from gravitational mass--therefore they can be measured separately or can be present independently.
The spacetime-curvature model or analogy you are referring to provides an example, in which mass is a quantity that is tied without a known mechanism to a conceptual curvature property, and is not necessarily identical to an inertial mass.

Doesn't general relativity show that gravity is a secondary result of mass affecting the curvature of spacetime, not a direct interaction between masses?

No, that was a descriptive analogy, not a mechanistic interpretation of how things really work--in some respects similar to Maxwell's equations, which model relationships between quantities but form no opinion as to the mechanisms. Note that so long as mechanism is in question, causality is up in the air, and in general it could not be inferred that a phenomenon is a "secondary result" of a model, equation, or analogy, since the precedence and "order of operations" in a physical sense is unknown.
