Here are two very good links to videos by famed and highly respected tennis coach Rick Macci discussing the UNIT TURN. The unit turn is one of the major changes in the modern pro tennis game. You don't have to be a pro to understand it either.
The role of the non dominant arm is to ensure that shoulder rotation comes first as you are running or pivoting. Look at these pictures of Federer and Nadal loading their shoulders on their running forehands:
You've seen it a hundred times, Federer and Nadal having their non dominant hand on the throat of their racquets as they are running. Is it faster to run with the arms closer to the body rather than having the non dominant arm dangling? Federer and Nadal seem to think so. Go out on the court and try this out. Judge for yourself.
How important is it to turn the shoulders instantly or to use bad form and arm the ball instead? Use both shoulders and you will make your life much easier. You will find that you will have more force on your groundstrokes while being more compact. Plus, your takeback will be much more precise and neat, and you will be able to judge your racquet face angle and swing path angle better.