Quote by ellipse
And maybe an unrelated question, is it better to think of curved spacetime as a model that explains the world very well, rather than the way the world actually is?

better to think of the world really being curved spacetime (the model used to talk about stellar collapse, observed kind of black holes, big bang, inflation scenarios, dark energy/accelerated expansion etc.)
I think it is is better to think of the "graviton" approximation as something that is typically useful in static almost linear situations. Haelfix indicates where the graviton approximation is applicable:
Quote by Haelfix
No, the two theories are equivalent, at least in the linearized form and to first order.

at least, that is, where there is not too great a concentration of matter, or too great fluctuation in curvature. Then you can approximate spacetime by a fixed 4D spacetime like e.g. minkowski flat (which is the spacetime associated with zero matter and zero curvature) and you superimpose on top of that a little ripple or perturbation. To FIRST ORDER that gives an approximation of the real thing, namely your basic curved spacetime.
you are approximating a dynamically curved thing by a static typically flat thing that you have added a little ripple or bump onto.
this is called a "perturbative" approach (take a fixed standardized typically flat thing and perturb it slightly, calculate first order effects)
conventional wisdom is that perturbative approach doesnt apply in regimes of high curvature and highly changeable geometry. It is a marvel (as Haelfix notes) that it works at all, even in the comparatively flat cases.
Would a graviton theory replace the curved geometry of general relativity? If gravitons are the cause of the gravitational force, does that mean space and time aren't really curved as Einstein thought?..

No, I dont think so. It is a marvel that the graviton picture works so well as a perturbative firstorder approximation of reality.
Personally I doubt it is helpful to think of "gravitational force" bending lightrays as they pass by the sun by means of clouds of gravitons whizzing back and forth. It is more like those are just the geodesics in our curved spacetime. Nor does it help me to think of a black hole attracting another black hole by the two singularities sending clouds of gravitons whizzing back and forth between themhow do the gravitons escape from one to get to the other etc etc.
So I consider gravitons as marvelous wonderful approximations in the static typically flat cases they apply, and as elegant mathematical constructs. But in answer to your question they are not the fundamental " cause of gravitational force" and NO it does not mean that "space and time aren't really curved as Einstein thought?"
I'm following current research in
nonperturbative quantum gravity, which has been making important progress recently. In the nonperturb. approaches to QG you do not use gravitons, you basically randomize the geometry of spacetime. so you have a quantum (curved) spacetime.
Nonperturbative quantum gravity typically means the kind of stuff at the Loops 05 conference
http://loops05.aei.mpg.de/
the graphics there can give some idea
if you do quantum gravity these days then most likely you think of spacetime as curvedonly the curvature is uncertain as quantum things usually are.