It's going to be difficult to bring a simple relation out of this, IMO, there are three levels of complexity to talk about:
1) Weight in a sling will inhibit your throwing speed as more force will be needed to accelerate the sling (and stone) to a given velocity. In this regard there is no advantage at all in making slings heavier or thicker than they need to be to not break (i.e. fishing line). However in reality some thickness in slings prevents tangling, is easier/faster to use and perhaps offers more control over the projectile. So fundamentally there is a trade off here and you need to decide where to draw the line. There is no *right* answer.
2) Talking about the projectile weight adds another layer of complication. For a start, the heavier the projectile the proportionally lesser the effect of the weight of the sling will have on your throw. If you throw an 800g stone, whether or not you have a 20g or 60g sling will not significantly change the amount of force required to accelerate that sling/stone system. If you are instead throwing a 40g stone then the sling becomes a much more significant portion of the system you are trying to accelerate, and therefore any change in mass of the sling will be felt proportionally much more so. To simplify the case; you won't notice 820g vs 860g, but you probably would notice 60g vs 100g.
(Note: I'm just adding these numbers together for a general illustration of the idea. In reality the mass of the sling would be 'felt' less than equivalent mass in the projectile. This is because mass on the sling is closer to the rotor centre and is thus moving slower which requires less energy)
A second point is that thicker slings bring benefits specifically for large/heavy projectiles. For example, they are less likely to tangle or twist round whilst spinning the sling. I've found also that stiffness around the pouch significantly improves ammo retention on LARGE projectiles. Conversely I've found thick, heavy slings can spoil release on lighter ammo.
3) The third layer of complication is the fact that this whole system is interacting with the immensely complex biomechanics of the human body. This is the part where you essentially shrug and give up hope of getting some nice simple answer. See the thread by Joe Meadmaker in which he measures the sling velocity on a wide range of heavy rocks.
https://slinging.org/forum/YaBB.pl?num=1628318591 Notice how in the graph I show, the energy of the throw increases as the weight of ammunition goes up. This is pretty much because Joe was throwing slightly less fast or even just as fast with ammo weight going way up or even doubling. This is something that doesn't make a whole load of sense if you just have a widget that puts energy into a projectile, but the human body is more complex than that. It appears that at least for Joe, using that specific technique in that specific ammo range he was getting a more efficient energy transfer as the ammo weight went up.
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All in all, If you wind back all the complexity I'd say yes- a lighter sling will perform better with lighter projectiles because it has a higher efficiency and the benefits of thicker heavier slings are not needed. On the other hand, a thicker heavier sling performs better with heavy/large ammo and the efficiency losses are less significant. However, exactly how light or heavy a sling should be for a given ammo size is a judgement call.