Wanted to keep a rolling record of games I have played recently that feature slings, as I find more I will try to remember to update it. I'll try to include steam links as well.
Northern Journey
https://store.steampowered.com/app/1639790/Northern_Journey/ weird Norwegian fantasy game, the first weapon you get is a sling and has infinite ammo with a powerup that can be found later to make the ammo flaming to increase damage. The amount of rotations actually correlates to the power of the shot, three rotations is max power and it can stay relevant the whole game (I managed to beat every boss except ones with specific mechanics that needed a special weapon with only the sling!) There's a lot of spiders, just as a heads up. It was my game of the year in 2021.
NEO Scavenger
https://store.steampowered.com/app/248860/NEO_Scavenger/ An incredibly frustrating and unforgiving turn based post apocalypse survival game. A sling can be crafted with a couple pieces of cloth and some string, both of which can be found by ripping up the hospital gown you start with. Simple and brutally effective, according to the weapon damage values on the wiki it's basically a handgun. I was constantly overburdened by carrying something like 40 large stones, I was a walking, incredibly aggressive, geology show.
A Plague Tale: Innocence.
https://store.steampowered.com/app/752590/A_Plague_Tale_Innocence/ Taking place in France in 1349 during the plague you are Amicia, a nobleman's daughter who has been trained by her father in use the sling. Great story, supposed to be a stealth game but when the sling is fully upgraded it becomes DOOM
Pathfinder Kingmaker
https://store.steampowered.com/app/640820/Pathfinder_Kingmaker__Enhanced_Plus_Ed... An isometric RPG based on first edition Pathfinder, there is a sling staff (take the exotic weapon proficiency at character creation, don't be a fool like me) but the sling itself was taken out in beta. Pathfinder Wrath of the Righteous -might- have one but I have not played it yet.
Far cry: Primal
https://store.steampowered.com/app/371660/Far_Cry_Primal/ Caveman adventures, there's a sling but you can't equip it per say and you have to do a bit of work and follow the main story to get to it. It passively replaces your ability to throw rocks to distract things but gives you the ability to absolutely brain anything (headshots are lethal) with infinite ammo helicopter sling. I basically stopped using the bow after I upgraded it.
I am going to include Mount and Blade: Warband which has a mod called "Bannerpage" that adds some quality of life stuff and that includes a sling and sling staff.
As I find things I will try to remember to update the list! If you know of others let me know.