If the grip prevents turning the wrists freely and makes you extend the arms to turn them at all, naturally your shoulders will respond and begin to help.
Take a club in your hands and try lifting it up over your right shoulder, keeping the left arm firmly extended throughout and bending the right elbow sufficiently to pull the club over until the shaft touches the shoulder, and you will find you have equal freedom with both the orthodox and the “Vardon grip.”
In the raising and lowering of your club you need freedom, but any grip which causes you to rely upon twisting your shoulders around to get the sweep of the club and makes it extremely difficult to get any sweep. At all in rolling the wrists is placing the burden of the work where it belongs. It will force you to turn the body to get around any distance with the shoulders. This is a very desirable result. This grip for the same reason makes it extremely difficult to finish the swing in any but a correct and complete turn of the body and shoulders, and naturally makes a good “follow through” imperative.
A very important advantage of the “Vardon grip” is the added firmness or control of the tendency of the club to twist in the hands as you, strike the ball. This is due to the fact that the left thumb down the shaft with the base of the right thumb pressing on it acts as a brake, so to speak, and saves many a shot struck too near the toe of the club.
If you will make the experiment of drawing up your club to the top of your swing very slowly, using the Vardon grip, in the same manner as you would use with the orthodox grip, you will find it extremely awkward in the first place and lacking in the feel of power in the next place.