I've been struggling to really understand the logic in these last few pages. Couldn't they just point the beam at nothing? Sure there's the argument that they might inadvertently hit something without intending to, but the exact same thing could happen by pointing it at Earth too. And if they somehow know that by pointing it at Earth would only hit Earth, then couldn't they use the same technique to find a path where nothing gets hit, not even Earth?

Yeah. The prostpects of hitting something while shooting through the galaxy at random are abysmally low. It's almost entirely empty. This is like being worried about hitting a pedestrian in a nearby city while firing a gun in the air in the countryside. . . Technically possible. At sectors away, any deviation will make them miss their target anyway. The beam would also have to be nigh-instantaneous FTL and not light-speed itself cause everything's in motion. Also no diffusion. I'd suggest the video 'metric paper' for some perspective on what exactly is in space.