Interesting idea, Joe. I would imagine that, ranking them purely by my limited atlatl and throwstick experience, the atlatl would be most accurate, sling second, and throwstick third. But that's just very general and anecdotal, there's a lot of variables to all three, and only one has fletchings to smooth out some variables. Of course I imagine that if you had a master for each respective weapon, they'd all perform about the same, perhaps with the atlatl having the tightest group. It's a close one though, and it may also depend on how accuracy is determined, i.e: by group, by time/attempts to hit a bullseye, by singular shot/throw, etc..
But for people with no experience whatsoever, I'd probably say the atlatl would be most accurate, throwstick next, and sling last. But in the long run throwsticks are hard to get accurate with, maybe moreso than slings. Ben has even had some issues throwing his own sticks - even at a couple paces... sorry that might be confidential information

...All jokes aside, he's a really good throwstick thrower.
But as mentioned I have very limited experience with all but the sling, and even then I'm not accurate