Some silliness in some posts above.
Threadless headsets allow cost reductions in bicycle manufacturing. One fork (cut to length by the LBS) works for all frame sizes, and never needs to go through a threading operation.
Quill stems are easier to set up and adjust to the rider. They also allow the handlebar to be higher (yes there are cludges to make threadless stems higher).
Headset/stem weight is a red herring. You really have to consider how much difference it will make in your ride. Generally, the honest answer is none.
As to twisting a handlebar so that either end shifts 2", well that's just not goinjg to happen unless something is broken. And, that might be the rider. There is no way, standing in front of a bike, that anyone can stablize the steerer tube enough to tell of motion is caused by the stem bending, or the more likely culprits of the whole bike moving, or worst case, the front wheel being soft and bending.
There is so much silliness of stiffness that is just so much marketing brainwashing.
So, either headset type and stem works just fine.