You would think so, until you try the stiffest setting in my truck. Going straight down a paved road and hitting a small bump or repaved strip causes my back end to go squirrely. It's very noticable. Only change was setting it to stiffest setting, and going back to middle hole eliminates it.
The reason is because when you hit a bump going straight, there are still forces at play that push you around a little bit to one side or the other, you never take a hit where the only force is 100% front to back. The more force you get that hits you off on the side, the more the swaybar matters.
So yes, the sway bar definitely controls axle hop, and there is such a thing as "too stiff". This was all in my lightly loaded bighorn, I plan to tow in a few weeks and will try stiffest setting to see if there is any improvement while towing.