Actually, Newtonian gravity can be added to QM and work perfectly well. It's GR gravity that doesn't work with QM, especially if you try to model very high curvature like you'd get near a black hole.
Sorry, but this is a pet peeve of mine: special relativity works perfectly well in accelerating frames of reference, as long as the spacetime remains flat (a Minkowski space[1]), for example when any curvature caused by gravity is small enough that you can ignore it.
What general relativity does different from special relativity is that it extends the concept of relativity from inertial frames of reference to all frames of reference, even accelerating ones. In the process, it also explains the reason why inertial mass and gravitational mass happen to be the same, by tying the gravitational interaction fundamentally to the concept of acceleration.