Bertholy does not talk about 'falling' but he does require some things that make it happen.
First what is necessary to 'fall':
1) Center of gravity never gets over your trail foot
2) Lead leg flexes allowing space for a foward and down movement of the body
Bertholy is specific that your weight should stay on the insides of your feet. And that on the backswing your hips should not move back. This satisfies condition one.
Bertholy is also specific that your lead knee should bend away from the target on the backswing and the downswing should begin with your lead knee rotating toward the target and bending toward the target line (flexing). This satisfies condition two.
Note that if your lead arm goes above horizontal on the backswing, the momentum of that movement will add to any movement toward the target in the 'fall'. Many pros have their lead arm nearly vertical at the top producing a considerable force moving their bodies toward the target. In general in my observation the higher the hands of the pro the more time between the forward movement 'fall' and the start of the hand down. This may be why a higher shoulder helped you but a high shoulder is not necessary only #1 and #2.
Even if your lead arm does not go above horizontal, starting the downswing with your lead knee per Bertholy and having the first move of the arms be the 'Claw' increasing the wrist cock will insure a forward body movement before you arms start down.
