I think we should begin by distinguishing among a set of different but related issues: what gets flattened? How best to flatten? What is the purpose of flattening? The general view as to the first and third questions is that the answer is that the purpose is to improve ball striking and that what gets flattened is the shaft. it lays down. So now the questions are: why would flattening the shaft in transition improve ball striking, which presumably requires improving impact: so the question really is why does flattening the shaft improve impact?
The answer is that on its own, flattening does not improve impact. So flattening was be matched with other movements or it must encourage or cause other movements that in fact improve impact. One familiar view is that laying the shaft down (some) encourages body rotation through the downswing which in turn lowers face closure rate and increases reliability of strike in several ways. But... flattening by simply letting gravity work on the club head to tip it down and thus behind the golfer in transition, opens the face relative to the arc; and an open club face invites a stall and a handsy release that is likely to increase the closure rate.
On its own, there are virtues to having the hands/arms in front of the body and the club behind and to the inside of the hands because from that position, if the face is slowly closing to become squarer to the arc as it approaches impact, the golfer can drive the bulk of the downswing through rotation, securing consistent low point and face control with relatively slower rates of clubface closure. Two obvious points then: the point of shallowing the shaft in transition is that it sets in motion a set of sequential movements that in principle lead to better ball striking. that's the first point. the second is that this can only work if the clubface is approaches the delivery position relatively square to the arc. That means that on its own shallowing is not enough to set this sequence into motion. that is one easy way of understanding the value of what Tyler calls the motorcycle move and why the motorcycle move and club shallowing moves work together to set the desired sequence in motion.
The second question from the beginning remains. Is there a preferable way of shallowing or laying the shaft down?
The two approaches that are discussed in the original post are not in fact two different ways of shallowing the shaft. They are both shallowing moves, but the first one discussed (the passive, leave the arms in the structure in which they are in at the top of the backswing and rotate) in fact aims to flatten the shaft; the second does (active, ulner deviation) does not in fact flatten the shaft, though it is a shallowing move with regard to the overall downswing. It shallows the swing plane but not by flattening or shallowing the path. In fact, it is possible, and for many instructors, GG and DD and also a fellow named Larry Cheung from Toronto among others to put both moves together. In fact all three recommend flattening the shaft in early transition by rotation, then immediately thereafter ulnar deviating (down not out) and then rotating through impact.
All allude to what Tyler refers to as the motorcycle drill under one or another description of it, but none emphasizes it as Tyler does. In contrast Tyler's conception of transition does not involve beginning with lower body rotation but with the kind of two inch move of the lower and sternum centers two inches forward (in the manner Mac O'Grady, for one, recommends, and which has been both a keystone of the MORAD method and deeply misunderstood by the world of Stack and Tilters). So, if I understand Tyler's view correctly, he is noncommittal as to how to flatten the shaft. You can do it either the passive way endorsed by GG) or through arm rotation (as endorsed say by Bradley Hughes). But what Tyler does recommend strongly is a move in transition and continuing into impact of 'twisting the shaft' to square up the face; he also favors ulnar deviation early at a point just about at the delivery position, perhaps helpfully seen as at the beginning of the wipe move..
So to get to the initial video, only the first move shallows or flattens the shaft. It is distinguishable from other ways of doing so in that it is passive, The shaft can be flattened actively by forearm rotation. But the point of flattening is that it is essential to an overall pattern or set of related patterns that improve ball striking by putting trunk rotation at the center of creating impact. Think of at least the following two as alternative patterns that begin with shaft shallowing or flattening to set up the moves that follow;
Pattern 1: Shallow passively by starting downswing with rotation, twist face square; ulnar deviate, continue rotating to p6.5 then push and jump to move the divot forward. GG
Pattern 2. Shallow actively by rotating forearms and shifting centers forward to inside of left heel, motorcycle/twist face square, as club approaches last parallel, ulner deviate, continue rotation and either wipe (extend arms diagonally) or just rotate.
Given the number of movements in each sequence, it is easy to imagine different ways of organizing the component movements.
Personally, I find that I am most comfortable with a variation of the second pattern. As an old GMachine guy who has worked a bit with Mac, I believe in moving the centers forward, while shallowing (my shallowing or flattening of the plane is nowhere near the extent you see in Matt Wolf or even more conventional of GG students). I don't think I ever get under the plane. My goal is to keep the hands behind the sternum, the head behind and inside the hands until hands are in front of trail leg. So shallowing is there only to ensure the club is not tipping out too soon to the target line. I also can accomplish this position only if I am attentive to actively lowering my arms, i.e. getting my left arm separated from my chest. As I approach this position I am rotating, ulner deviating and extending my arms. Variations of this is what I try to teach my clients depending on how willing to work they are, how athletic they are and how willing they are to put up with shitty shots until it takes hold.
In my view shallowing is not a magic move; it is a way of insuring that the clubhead is inside the hands which are themselves behind the sternum when its time to ulnar deviate and add arm extension to rotation. And if you flatten the shaft you have to add a move to compensate for opening the face.
Finally, to get back to the original question: in my honest and humble opinion, the ulnar deviation move is a shallowing move but it does not shallow the shaft. to my mind the shaft is shallowed only when the butt end of it points farther beyond place on the ground that it did prior to the top of the backswing; typically it will point to some degree outside the ball or target line.]
I hope this is helpful but I worry that it's not. In any case please excuse my rambling.