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.
Drive drywall screws through the drywall on either side of the hole and the wood with a power drill. Cut a patch of drywall to fit in the hole and screw it to the wood. Place fiberglass tape over the seams and spread mastic, or your selected tile adhesive, over the fiberglass tape with a putty knife.