I may be wrong, but either arm can pull the arrow from the quiver. Just avoid the right arm paddlewheel action and monitor the pull.
A right-handed archer holds the bow in in left hand. He does not swap hands or drop the bow to reload. The English longbowmen carried approximately 72 arrows each in a battle, which they could fire off in minutes at their maximum rate of fire. Speed was essential so they would not be swapping hands during a battle to pull another arrow. They would have pulled the arrow with their right hand whether the arrows were in a quiver or not.