The problems with a sling include:
1. You pretty much have to be in the open and exposed to use it.
2. It takes some room to twirl the sling and throw the stone / shot.
3. It is slow firing. You have to load the sling then spin it one or more times before release.
4. If you are using random rocks as the missile these will have a bigger dispersion pattern due to their different sizes and weights. Missiles that are made to the same size and weight will be less problematic.
Its an interesting skill but one with limited practicality
1: only partially true- and only for a hand sling - you have to be (sling+arm) length back from partial cover, but can be in about 2/3 covered positions. (Celtic slingers fired from palisades, for example.)
For a staff-sling, totally untrue. I've fired for effect from 75% covered positions.
2: true for a hand sling. Mostly False for a staff sling... you need space behind and above equal to staff+sling length.
3: Actually, false. a competent slinger can get "low" power shots with 0.5 rotations... A staff sling only rotates about 120°...
4: quite true, but not nearly as bad as you imply. But almost any military ammo can be loaded into the sling for good effect. Got a pile of 7mm ammo, and a 5mm rifle? Pop the bullets off with your leatherman, and use them. If you have time and epoxy, glue two slugs butt-to-butt. (better ballistics).
Typical riverine small gravel stone is highly accurate.
I know a celtic warfare recreator. He's fired random rocks with half a rotate 50yd with hit-a-man accuracy.
Also, the israeli army has found that west bank arab slingers are still a major threat... a bit of leather and some cord, and any yahoo can wound or kill a soldier with just a little practice.