Hi KC,
Since you led me to make a fool of myself on an earlier exercise you posted here, I am always wary about making comments to your posts which are at first sight counter intuitive. However, against my better judgement, I feel like jumping in anyway.
You have already made the mental simplification that the tree is a massless cylinder with all the mass collected at the centre of gravity (CG). Could I for the sake of further simplification propose that we reduce the length of the log to just a slice and then take a plank from the slice going top to bottom - in other words, we now have a pole, stuck in the ground one end (friction of the log against the earth we assume prevents slipping so at any one point of contact, the contact is static), and with a rope tied around the top end (again we assume that the rope does not slip over the surface of the log). In line with your previous assumption we will assume that the mass of the pole resides in its CG in the mid point of the pole.
Now if we pull on the rope and move the top of the pole forward by a foot, the the centre of the pole moves forward by 6" : mechanical advantage (MA)= 2.
Of course, no matter what length the pole is, the middle only moves forward by half the movement of the pulled end, so the MA is always 2 irrespective of the length or mass of the pole.
The reference to using a cant hook is I believe a change of circumstances because using it, you are changing the pulling radius while leaving the rolling radius unchanged.
So for a log of 2ft diameter (a pole 2ft long), pulling the top forward by a foot moves the CG sitting 1 ft above the ground forward by 6" (MA = 2). However if I stick a 4ft cant pole onto the top of the log, the Cg is still only a foot above the floor, but the pulling point is now 6ft away from the ground. Now, if I move the top of the cant pole forward by a foot, the CG only moves forward by 2", so yes I have improved the MA to 6 but only because I made the pulling radius larger than the rolling radius. In a log the CG always stays in the middle irrespective of the logs diameter, so the pulling radius and the rolling radius stay in constant ratio irrespective if density.
However, despite the amazing impact of this image of human ingenuity from a time gone by, for me there is another issue at question. The real issue is that the Parbuckle is a rope device (machine) and not a knot, much like a hammock is a device made up using many knots but is not a knot itself, so the Parbuckle is a device made using virtually no knots (and indeed the load itself becoming part of the machine).
Although I think of knots as being 'rope machines', the important distinction for me is that in a knot, this machine bears upon itself to achieve its functionality. Even in hitches where some 'third party component' is an essential part of the functioning structure, at some point the rope still bears upon itself to achieve the functionality of the knot. Inevitably, such a description leaves out elements of importance to rope use such as the almost vital 'Round Turn' - I class the round turn and the Parbuckle as rope tools not rope knots.