I think that when we look at the topic of hunting with a sling we become somewhat blinded by our modern concepts of hunting as individuals, rather than as part of a coordinated group in a social setting.  While I think it likely that many birds and foul can and have been taken by the sling, including while on the wing, I think it possible our ancient ancestors could have used it to take much bigger game.

I think we have enough evidence for the herding of wild game over cliffs or into traps throughout our history to consider it real.  I don't think they were necessarily common, but maybe for those times of year that large groups would gather - regardless, I seem to remember from the depths of somewhere an old illustration of people waving blankets or some such thing to get the herds stampeding, but to be honest I don't think that's going to get a mastodon or bison moving.

What I do think might work would be the buzzing sting of a few dozen slung stones.  It's fairly well known that some herders still use slings today, though with a very different intent - they generally lob a stone to make a noise, this envisioned mammoth drive would be different, with an intent more to push and frenzy.  In this fashion a slung stone, if it hits, isn't intended so much to cause enough traumatic damage to kill or immobilize, as to confuse and to harry, and even close shots can be fairly terrifying - certainly enough to convince you to move elsewhere in a hurry.  Maybe even enough that one might run blindly over a cliff.  This may even be the origin of herding with slings itself.  Another factor to consider might be the relative scarcity of good timber in a predominantly tundra environment, especially as opposed to fiber for cordage.  Smaller groups with fewer needs could have focused on a single individual, running their quarry between groups until they collapse from exhaustion.

Of course this seems doomed to be forever academic.  I'm not sure how one would prove it one way or the other, as we seem to have a generally hard time recognizing and respecting the sling in antiquities research even under the best of circumstances, let alone under the conditions and ages I'm proposing.  That being said, I think it possible and even likely that the sling would have been integral to these sorts of drives.  Of course I'm not aware of any documented instances of sling being used in this way, so the whole thing might just be armchair fantasy.

That was a lot, but it's pared down and informal - no research included (but those rabbit holes are out there and easily accessible), just spitballing ideas, I only hope I made half as much sense as what I'm going for without rambling too much, and that what ideas are there are worth following through to the end (if you've made it this far).

Cheers.