The slingers are the first tier ranged unit that all the eastern people get. I have been playing as Pontus and not sure if they are the exact same as from Greece and further west. The slingers have longer range than archers, which really irks me, and apparently laugh in the face of hoplite or shield formation armored units. They die like nothing if you reach them it is just amazing what they do in a couple volleys sometimes. There is definitely no waiting out some ranged units like I did in Rome 1 with Romans or Greek hoplites.
They should have. Remember, these weren't English Longbows or composite bows (and there's still theories that slings would have equal effective range to English Longbows if slung by somebody grown up in a slinger culture - but I take those theories with a large grain of salt). They should be effective against heavy troops as well, seeming as both stones and shaped bullets travel with enough force that even if they do not break the armour (which was well within their power) they can still break the mushy meat-and-bone-stuff on the other side of it.
Slings were basically the English Longbows of the Classical Age. They took a lifetime to master, but they dominated the battlefield. The reason the Greeks and Romans thought of it as a "dishonourable weapon" was, in my uneducated opinion, just because they were sour because they risked getting outclassed by a bunch of goat-fucking mountain-herders every time they looked east.
That said, that's pretty much the opposite of my experience. I'm having trouble killing anything with slingers.