As per the comic, you can't orbit faster without increasing the orbital altitude. You can't just stay at the same altitude and "go faster". Orbital motion doesn't work like that. Also, moving large ships is very often a slow and imprecise task.
To give a nautical example, it's not unusual for a fully loaded modern supertanker to take 3~8 km to complete a braking maneuver from its normal cruising speed. This means if you're cruising along and decide you need to stop, and you throw your engines in full reverse, it's going to take 14 minutes or more to stop.
And while some large ships have maneuvering thrusters or bow/stern thrusters that help it turn without moving forward, they're mainly used for docking. For normal turns out at sea, a large tanker usually has a turn radius of over 3km.
A space station like the Death Star is going to have a lot more momentum and even less agility. Plus, it's much harder (takes more energy and time) to brake in space. So simply waiting for the orbit of the Death Star to put itself in position for the shot does make sense.
Even in space, there are gains and tradeoffs when it comes to size.