Yes, the left shoulder does travel up plane to get the right shoulder downplane. Brian posted a clip demonstrating this, (which looked like it was filmed with a hidden camera), and briefly discusses this in his Flipper video and vigorously discusses this in a past forum discussion.
Homer said swingers should pull with anything. My instrument of choice is the left shoulder. Personally, pulling the left shoulder up plane on the downstroke does indeed get the right shoulder moving down the turned shoulder plane and creates that requisite trailing, lagging feeling and consequent inside delivery path and throwout action.
The left shoulder doesn't power the stroke. To me, monitoring and thinking about the left shoulder is another pathway to understanding and learning about the proper motion of the right shoulder on the turned shoulder plane. A mechanical equivalent.
Taken a step further, I find monitoring,Look,Look,Look, the left shoulder on the backswing helps me achieve a relatively flat right shoulder turn, even with a right forearm pickup.
I used to do the opposite, and probably still do at times. My shoulders rotated too steeply on the backswing, and not steep enough on the downswing. In my case, only thinking about the right shoulder wasn't enough. Only half the picture.