It amuses me that there are lots, and lots, and lots, of threads on various forums where people foam at the mouth "you can't load a hand crossbow one-handed" yet I think a sling would be harder to load one-handed. Also, rules-as-written, a level 20 fighter can fire a sling four times a round while only firing a hand crossbow once.
My experience with a sling is that it is not just point-and-shoot - it takes a few seconds to wind up to a good speed before you can loose the missile.
It is also a *lot* harder to learn to aim well with a sling than a hand crossbow, yet in the game the sling is simple and the crossbow martial.
It's true that a sling requires a minimal amount of dexterity with a second hand to load, but it's easily accomplished even while holding a shield, as this guy demonstrates:
http://youtu.be/gXiUDJRgiUc
Slipping a stone into a cup is perhaps a little more cumbersome than nocking an arrow, but it's nothing like loading a crossbow, which requires a least one limb to brace the crossbow and another to apply considerable force to a lever or crank, and often a third or even fourth limb gets involved at some point.
Crucially, once the stone is in place, actually firing a sling can be accomplished with a single hand. And whirling it around for a few seconds is not necessary. Many slingers will give it a revolution or two to seat the stone (takes less than one second), but a skilled slinger can launch it with a single revolution, like this guy:
http://youtu.be/fWHV2WYdfTI
You're right about accuracy, though. Crossbow is far more accurate even with basically no training. The being said, the PHB probably underestimates the effective range of a sling in skilled hands, which was said to out range bows on some ancient battlefields.
I assume the fact that some crossbows are martial reflects the strength and technique required to draw them, just as the longbow is martial even though it's based on the same principles as the shortbow. And yeah, game balance, I guess.