If the Moon disintegrated as in Seveneves, certainly not all of its mass would end up hitting the Earth, but some of it would, and it'd only take a small fraction to produce catastrophic results.
In general, if something in orbit breaks into pieces moving away from each other, then by definition neither piece is still in the original orbit, because (due to conservation of momentum) neither piece has the original combination of velocity and altitude. One piece will be spiralling toward Earth, the other away from it, at a rate determined by how hard the object broke apart. As the fragments of the Moon collide and break each other into smaller and smaller bits, they will spread into a wider and wider distribution of orbits, some of which will eventually pass within the Earth's atmosphere, while others escape into deep space.
To put it another way, the Moon disintegrates, and becomes an expanding debris cloud like the Death Star exploding. That debris cloud is a disc centered on where the Moon's center of mass used to be (more or less). Sooner or later, the disc will expand to touch the Earth's atmosphere, and at that point the dust / rocks / huge boulders that make up the cloud will start falling out of the sky. Even if most of it burns up, this could easily heat up the atmosphere by tens or hundreds of degrees (which is what Stephenson calls the "white sky").