Ever see someone with those rounded, forward shoulders? Yeah, that affects their scapular coordination and muscle activity and may cause shoulder impingement. By better positioning the shoulder complex and particularly the scapula which is where the rotator cuff actually attaches, you may improve activity of the rotator cuff (which acts to provide active stability to the shoulder explained here) and in turn cultivate better stability and function of the shoulder girdle.
The guy in the video above (don’t worry, what I lack in genetic height is made up for with pristine movement) also has a history of an AC joint (acromioclavicular joint) sprain in which there is a separation of the clavicle from the acromion process of the shoulder blade. Because the AC joint needs to move to accomodate scapulohumeral position, this further disrupts the mechanics at the shoulder and underscores the need for improved positioning. You will see poor scapulohumeral positioning in those whose elbows fly out to the side during pressing/pushing which is typically indicative of a lack of internal rotation of the shoulder. You can see on the push-up test and re-test in the video that he is better able to maintain his elbows close to his side, indicating that the humerus is more towards the back of the socket allowing better expression of motor control instead of relying on non-contractile tissues like his joint capsule and labrum causing nasty dysfunction and chronic pain in his AC joint.
Try this technique for a few minutes (2 minutes at minimum) to improve your push-up or possibly bench press. I tend to like this technique for push-ups due to the kinetic chain demands: we need to mobilize in a closed-chain to match the position of restriction during push-ups. Remember, the goal is to train the movement so we need to treat the movement. If you are doing a lot of push-ups (and you better be) try this out.
P.S. You can vary the load/intensity by making your body more parallel to the ground. Use a bench to increase the load of the stretch compared to the wall.