Consider the firing methods of the two weapons.
The bow is: nock your projectile on the string, point, draw, release.
The sling is nest your load in a pocket, swing in rotational motion, then release. Timing that release point takes skill. Do it at the wrong moment, and you could even fire backwards.
And the bigger the payload, the longer the sling, and the more space it requires to use. I’ve seen ones big enough to launch half-bricks that were around 2’ long.
Huh. I used to make slings when I was a kid (not much else to do on the farm) and I always thought it was pretty simple. Arrows were easier to aim accurately though. Regardless, I don't see that it really matters. Slings are simple and cheap which is why I associate them to rural commoners, shepherds and halflings in my head.