The weight is back of center when the stride foot lands because of the coil and because the hands are back, not because of a big shift back. It is very much like the move you make when you go to throw a baseball. 

There are many different styles to getting into a good position to hit. Some players coil into their rear hip more than others. Some players have a high leg kick, while some players toe-tap or do not take much of a stride at all. No matter how they do it, when the stride foot lands, all great hitters are in an athletic and attacking posture 

The first move is in, not out. This inward move sets up a clean connected turn of the backside through the ball. It allows the bat to stay inside and release from back to front rather than out around the body. Remember, the swing is a backside drive and delivery, not a front shoulder pull.

Wherever the knob goes, the barrel will follow. The bat should stay inside the hands and above the rear forearm on its way into the hitting zone. The front bicep stays in with the chest, the distance between the elbows stays the same, and the hands stay near the armpit as the backside turns through. The result is a tight downward arc that flattens and naturally turns up as it rips through the zone... down behind the ball, then slightly up through the ball.  

The downward path of the bat will only flatten and turn up if your weight is balanced over your rear leg when you launch your swing. If you get out over your front foot before "go", the bat will start down and stay down, and it will never be on plane with the pitch.  

Because pitches come in at a downward angle, a tight path that changes from down to slightly up puts the barrel in a direct line with the baseball for its entire path and maximizes the opportunity for solid impact.  In the clip below, the dashed red line is the trajectory of the incoming pitch. Watch how his head stays still and behind the ball after the stride foot lands. Watch how early and closely his swing lines up with the pitch. He does not need perfect timing to hit the baseball hard. If he is a little early or a little late, the barrel is still going to be behind the ball, and he can still square it up with authority. This is the definition of hitting in a big zone. 

The spine angle is maintained throughout the swing. You want the feeling of swinging under your chin and front shoulder to ensure a backside delivery. Forward bend at the waist at the beginning turns into a side bend over the plate at the end of the swing. 

The rear shoulder comes through at a lower level than the front shoulder, but how much lower depends on pitch location. The lower the pitch, the lower the back shoulder should be. The higher the pitch, the higher the back shoulder. On the best swings, the angle of the bat and the angle of the shoulders will be nearly the same at impact.