I fought the Galatikoi Tindanotae, the Galatian version of the nude, mad drugged up barbarians with mostly slinger armies. The key is in flanking. Even against phalangites the armor bonuses from the front are simply overpowering. You need height or mobility. Easiest way to kill elite units with slingers is to leave one or two firing at them, make sure to disable skirmish mode. Then disable skirmish mode from the rest and have them flank on both sides. The elites will charge those units with disabled skirmish modes. Once your flanking slingers are on both sides and to the back of the charging elites, reactivate skirmish mode for your bait and watch the crossfire oblitirate the enemy. Works for Argyraspidai, Thorakitai Argyraspidai, Galatikoi Tindanotae, Hypaspistai, Hetaroi, Hellenikon Katapractoi etc. etc.

edit. Forgot to mention. Allways focus your fire on a single enemy unit. Minimum of four slingers per enemy unit, not counting whoever you got as bait. The Celtic Slingers are even better than the eastern ones due to their freakishly long range and almost inexhaustable supply of ammo.
That is quite effective even against the uber tank type guys as you mentioned. My personal tactic is to get slingers on opposite sides of a unit by using one slinger as bait and having one eventually slip behind it. That way, whichever way it turns it won't be able to respond effectively.

I should have mentioned my post to say that I was fighting three armies of naked fanatics at a river for fun. So, yeah if you outnumber them with slingers, you will win.