I started out exclusively with single-strand(floppy) slings, and stayed that way for almost two years before making any significant attempt to use braided(stiff) slings instead. My accuracy was plenty good with single-strand, but I did run into some issues with tangling cords occasionally, mostly in the beginning when my technique was bad.
After switching for almost a year now, I feel I do prefer braided/stiff slings and that they do give a slightly better level of control and consistency. However, I don't believe that difference would make a large difference in final accuracy, and that it's mostly just preference and what you are most familiar with.
The main thing I found is that your technique has to be quite different with single-strand slings than with stiff slings becauase single-strand slings lag so far behind your arm/hand motions compared to stiff slings. Stiff slings will follow your rotations almost exactly, whereas single-strand slings will take some time to "catch up". Acroballistics has a great demonstration on this in his torsion video here(the whole video is worth watching):
https://youtu.be/vOuRwrfA7GQ?t=133I think the main reason that most people who are used to stiff slings get inconsistent release with single-strand slings is simply due to this difference in technique, and they haven't learned to compensate yet. You have to apply the force in different places during your powerstroke or you will "overtake" the single-strand sling and it will tangle, reduce the final release speed, or release at an unexpected angle. In my own experience after a few tens of thousands of throws, my accuracy with stiff slings is just about the same as it was with single-strand slings. But for the first few thousand throws, it took quite a bit of adjustment. And now if I switch back to a single-strand sling, I struggle for a while before I am able to properly adjust.
This applies both with balls and with stones, but depends a lot on weight distribution of the projectile - the problem gets worse as the weight increases and/or gets further from the center of torsional rotation of the pouch. Again, the Acroballistics video has great information on this.
Separate from the stiffness of the sling, I am curious if maybe the increased weight of the pouch gives better proprioception since you have more feedback from the pouch during your throw. I'll have to give that a try.
Side-note here - while both Primal and Pan slings look like single-strand slings, their stiffness is much greater than it appears and is closer to a braided sling than a single-strand. They both pride themselves on their cords having a long internal splice into itself which doubles up the cord density/stiffness quite a bit. If you compare either one to a standard single-strand sling of paracord or dyneema, you will feel the difference immediately. Additionally, Primal waxes his slings aggressively, and Pan recommends doing the same(although isn't waxed out of the box), which also great increases stiffness even after breaking in. These two factors give them much better torsional control, which both of them actually include prominently in their "sling features" on their websites and various videos on their channels. I've made and used a lot of plain 2mm dyneema slings, and believe me when I say they behave nothing like a Primal sling with that doubled-up internal splice.
Mersa I've got no explanation for

dude is just a beast with a single-strand sling - back to what I said earlier about it mostly coming down to preference and familiarity.