Toucan,
In trying to adapt your analogy, I see the advantage in using the arms to help in a straight upward movement by using them to pull on the rope in augmenting the muscles of the legs.
You're right, there'd be a big difference.
In actually trying to convert that to real walking, I'm having a hard time making the transition. Rope analogy assumes (to me, at least) totally different momentum mechanics in that with the rope/squats you'd be using roughly centered (presumably) upward muscular efforts (both arms, center of forces & direction being equal left to right & straight up) that don't translate to the types of off-center (one-sided) un-equal (one-armed) non-straight up (angled) mechanics involved in walking with a staff.
I can see some of the "pulling" involved in walking up an incline (but my recollection is that it was usually miniscule), no real value on a level surface, and mostly in the way going down an inclined trail.
I'm extrapolating, and even though I've never backpacked, I suppose I can see the value in having a third point of contact in some situations as another balance aid when carrying an additional load on the upper body.
I'm mostly looking at a stick (or two, in whatever configuration) in conjunction with woodswalking without a load, and I'm not seeing the reduction in weight percentages on the hips & legs commonly quoted.
I'm admittedly outside the norm on much of my outlook on life in general, and I'm not knocking those who use sticks/staffs/poles, just not seeing the appeal or the utility.
For me, just about anywhere but over running water, mostly just more weight.

Over running water, worth its weight in gold (if long enough), but that may be a matter of percentages in a different direction. If you do cross a lot of creeks, that may outweigh the nuisance of toting the stick (or pole) (or poles) through the territory where it doesn't offer much advantage.
Percentages ran the other way in my case, I didn't cross enough water to justify the weight on the rest of the outing.
Ramm,
Wasn't confusing two poles vs a staff, just jumping into the related thread hoping for enlightenment.
But & however, outside of snow, for me two poles would be much more of a hindrance. Two long extensions to coordinate, both hands tied up.
For a guy with a level of coordination that frequently takes several tries to get his shoelaces tied, I'd probably hurt myself with them....
Denis