Given the size and weight of the ammo. I'm guessing what they have there is staff sling ammo.
Plus the length of the slings depicted is most likely more for artistic licence that realism.
For repetitive barrage slinging with large rocks you'd either use a fairly long sling (so the sling does the work) or use astaff sling.
The sling in the display case looks like it's been made to the same size as those on the tablet.
So - personally - i'd say not very realistic.
I do like the idea of pistol shot slings though.
That noise followed by the angry bee swarm noise of a sling stone apporaching would put the wind up even the bravest soldier.
And with moulded sling bullets you don't see them coming just hear them (And I do know that from first hand experience