(Ready for a long one?  This should be fun!)
I know this has never been discussed ever before on this forum (sic), so let's talk about the relationships among sling length, sling rotation, shot speed, and accuracy... only let's throw a little biomechanics in for good measure 

I am developing a "simple" physics model for the sling. Let's called it the "NOOC simplified sling model, version Zero" (or "NSSM-0" for short). What's the purpose of [any] model? A model is supposed to be a simplified representation of reality that has predictive powers within the bounds of the model's validity.  In other words, if the model cannot predict real-world behaviors or effects in some way, the model is useless, but if the model does not entirely reflect reality, that just means you found one of the boundaries of the model's validity.

Here are the rules of this model:
-Model is based on a 2D "world" with a top-down view. There is no gravity or wind resistance.
- The sling has no mass. It only has length.
- Targets and sling bullets are represented by 2D shapes in a 2-Dimensional space.  A "hit" happens when a target shape and bullet shape intersect with one another.
- The sling bullet's mass is irrelevant. The bullet has a speed, but no energy or inertia. This means there are no terminal ballistics. There is no transfer of energy or momentum between objects. Either the bullet hits the target or it misses. That's it. It doesn't hit hard or soft... It just hits (or misses).
-Simplified body motion: the hand does not move. The sling swings around the hand in a perfect circle at a controlled speed. The slinger's "style" either controls the rotational speed or the linear speed of the shot
-No power stroke, and no internal ballistics effects (spin, etc.).
- The pouch-opening effects are modeled as a fixed time delay. When a throw is initiated, the sling continues to rotate for a specified period of time, and then the massless sling magically disappears. The ammo continues in a straight line in the same direction at the same speed as soon as the sling disappears.

Biologically-Induced Timing Errors
Now let's add some biomechanics constraints on top of this sling physics model. Let's first assume there is a fixed timing error introduced by the human body. When the slinger intends to release a shot, they might be early or late within some small statistical time window. We will call this the biomechanical time-error window (BTW).

We can also assume there is a biologically-induced anglular error, but for simplicity (for now), let's just pretend that these don't exist and that timing is the dominant form of biological error and explore the angular error idea another day.

"Styles"
Next we are going to introduce two movement models (styles):
1. Constant angular velocity. This model assumes that the slinger is swinging the sling around in a circle at a controlled constant rate of rotation (measured in degrees/second, radians/second, or rotations per second).  The rotation rate is the same whether the sling is long or short. As the sling gets longer, the shots go faster.
2. Constant linear velocity: In this style, the slinger is controlling the speed of the shot instead of the rate of rotation around the hand.  The shot speed will be the same whether the sling is long or short. This means that as the sling gets longer, the sling rotates slower around the hand.

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Now... definitions are [mostly] done, so let's talk about how the model predicts sling behavior:
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Accuracy and timing:
An accurate shot is a shot that launches a bullet in such a way that it intersects with a target.  Since the bullet swings around the hand in a circle prior to the shot, accuracy and timing are synonymous.  To release a bullet at the correct angle, one must release the bullet at the correct time.  Further, there is a "hit time window" (HTW) within which a release will result in a "hit". If the shot releases outside of the HTW, the shot will be a miss.

How does this hit time window relate to the slinger's control of a shot?  Well, the HTW is inversely proportional to how fast the slinger spins the sling.  Faster spinning means smaller time window to get a hit. Spin rate directly drives how fast the bullet flies towards the target, so based on HTW, we can conclude that slow shots will be more accurate than fast shots.

What about the sling pouch delay? Since it is fixed, a fast-opening sling and a slow-opening sling will have the same accuracy, but the slinger must release the shot a little earlier with a slow-opening sling.

Now we can define a "timing error" as the difference between the moment the bullet must leave the sling to hit the center of the target and the actual moment when the bullet leaves the sling, and the ideal release time is going to be a function of the pouch delay.

Since we assume a fixed error BTW, this model results in a fundamental limit to how accurate a slinger is capable of becoming regardless of how much they train or practice. This accuracy limit is defined by the relationship between BTW and HTW.  If BTW>HTW, then it will be impossible for the slinger to hit the target 100% of the time regardless of their skill level. If BTW<HTW, then BTW can be ignored, because it cannot cause a miss on its own.  Under these conditions, misses must be caused by some other error. Presumably, these "other errors" may be corrected through practice and training, so there is still an opportunity for the slinger to improve. 

So what can you do if BTW>HTW? Since BTW is fixed, but HTW is inversely proportional to the rate of rotation... slow down to improve accuracy!  according to this model, the most accurate shot is the slowest one.  (Of course this model completely ignores the vertical dimension of a shot, so this model breaks from reality in a 3D world when you slow down too much)

Accuracy and Geometry
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Accuracy, Sling Length, and Target Distance
Assume the slinger is standing directly facing the target with the center of the target lined up with the center of the slinger's head. Draw an imaginary line from the slinger's nose to the center of the target. If the slinger fires straight ahead, the bullet will travel in a straight line that is parallel to the "nose line," but the path of the bullet is offset by the distance between the slinger's nose and the sling pouch at the time of release. Thus for a given target width and slinger distance from target, there is some offset angle at which the slinger must throw to hit the center of the target.  As the target gets farther away, the offset angle gets smaller.  As the target gets closer, the offset angle between the sling pouch and the slinger's eyes will increase. As the sling gets longer, the offset also increases.  Thus a short sling will have less of an angular offset and will require less triangulation than a long sling.
Now we can define "triangulation error" as the difference between the ideal release angle (angle at which the center of the bullet would hit the exact center of the target) and the intended release angle (the angle the slinger thinks they need to release at to hit the center).

Biomechanics and Accuracy versus sling length:
There are only two types of errors in NSSM-0: timing errors and triangulation errors... however these errors can be caused by many things. We already talked about BTW as a fundamental limit to accuracy for a given slinging speed, but there are others as well. For example, when a slinger is used to slinging with a sling of a particular length of sling, then there's a pretty good chance their accuracy will get worse if they start using a different sling length with the same ammunition (based on personal observation). Let's look at our two biomechanical movement models and assess how changes in sling length can impact timing and triangulation accuracy:

In the constant angular velocity style, the length of the sling does not affect the timing of the shot except for a small shift due to the change in triangulation.  For targets that are far away, the triangulation differences are negligible, and the size of the hit window HTW is basically independent of the sling length. As the rate of rotation increases, the hit window decreases. As discussed before, slow shots will be more accurate than fast shots based purely on the fact that there is a larger time window in which to hit the target. But since the rate of rotation is fixed, the slinger can compensate for the slow rotational rate by using a longer sling to get more "power" (shot speed)

In the constant linear velocity movement style,  the shot exits the sling at the same speed regardless of the length.  This means that, for a given shot speed, shorter slings must rotate faster than longer slings.  Thus the power is fixed, but time window for a "hit" is longer for a long sling and shorter for a short sling, so long slings should be more accurate than short ones.  Of all the things I've said so far, this conclusion is the most likely one to be controversial. We can temper the controversy a bit by talking about situations where the target is close to the slinger, but I think I've said enough for now, so I'm just going to stop here and ask for people to critique the model and/or give constructive feedback.

What do you think? What other features should I add to this model? The obvious next thing to add to this would be the concept of mass for both the sling and the sling bullet... because then we can have nice discussions about "power" in addition to all the debates this model should generate regarding length vs accuracy 

That's really interesting, love it!
I think the abstraction of BTW and HTW is really useful and accurate, and it's independent of internal ballistics.

For new features, I'd love to see double-pendulum mechanics.
You could model the slinger as a non-hookean torsion spring which drives the pendulums.
Then we could try out different relationships between spring force and deflection, and maybe learn how to optimize our muscle engagement.
Double pendulums are non-linear in general. But I assume that small variations in the parameters (i.e. the angle at which the power strokes begins, at which the sling is released, and how force is applied) should have a small impact on the power and the release angle. That would explain how the sling can be used accurately, and in turn help us identify which parameters are particularly crucial for accuracy.
Maybe similar research already exists for trebuches?

Surprisingly...my hit rate was about what it is in real life...1 in 5 to 1 in 10.

It is pretty fun.

I wonder if there would be much benefit to trying to add power-stroke physics into it. We could test a few theories about the radius of curvature during the power-stroke affecting the timing error, potentially. 

I agree with kairo on the abstraction of the timing errors being a very useful thought experiment.