It sounds like you may be referring to what I call the “tracking” mechanic - the physical source of the motion that moves the pick from one string to another. To make matters more confusing, this can sometimes be the same as the picking motion, and sometimes not!
In the case of sweeping, what you’re describing as a ‘sawing’ motion is what I think of shoulder tracking. The entire arm pushes through the strings. I think of this as a tracking motion because if you were to stop at a particular string, and play a tremolo, you probably wouldn’t use shoulder motion - you’d use a picking motion that originates at one of usual places - elbow, forearm, wrist, and so on.
To take another extreme example, if you were anchor the wrist firmly on the bridge, and only allow the the hand to move at the wrist joint, then your picking motion and tracking motion would be the same. Picking back and forth on a single string would be wrist motion. Moving to a new string would also be wrist motion.
In actual practice, players blend all of these movements. In our interview with Michael Angelo Batio this year we talked about that, starting around 55 minute mark:
When Mike sweeps, he appears to use a shoulder / elbow tracking type movement, with some finger orientation to control the pickslant based on the direction he’s sweeping. The finger changes appear deliberate, and are probably not simply a consequence of the pick pushing on the strings.
If you’re asking does grip influence all this, sure, it does. Can I use a shoulder tracking motion for sweeping, while using a trailing edge grip? Probably. I’m not sure that one precludes the other. I suppose we can try all the combinations. But given all the possibilities, another way to proceed here is to find players who do something, and do it successfully, and mark that particular combination as ‘possible’.