Duke William of Normandy won the battle because was well prepared and had a good army. They prepared carefully for the battle. The Normans had knights on horseback who were skilful fighters.
A utility knife, metal ruler, and a hammer can be faster than trying to carefully saw an opening. Score the drywall a couple times till you are most of the way through, then just knock a hole with the hammer and break the drywall back at the score lines.
Instead of using a circle cutter, trace the round hole with an extra box (Photo 2). If you don't have a box or a compass, sketch the curve freehand. You'll be surprised how accurate you can make it. Then cut the hole with a drywall saw (Photo 3).