I could be wrong. But I think :

1) All peebles revolve around their center of gravity.
2) For all peebles, the path of the center of gravity is right.

And more,

3) A flat peeble behaves like a fin in the air.
4) Aerodynamic peeble behaves like a football (non-amecican) ball with a drived path (Magnus effect).

just my idea.

This video from IronGoober is definitely worth checking out: https://youtu.be/ESrvvBFcrWM.

I don't know if it's entirely applicable because he's slinging a ball, not a flat stone.  But it might introduce some ideas to you.

The rounder, the heavier your projectile is, the less the magnus effect will pull it off target. Most important thing is putting rifle-spin onto the projectile if you want to hit things past 30m.
Flat stones curve a lot to the right when you throw them at medium range, but they also release a bit later than rounded ones, so to hit the target, you have to be aware they can cancel each other out to some extent.

If you want to curve rocks choose a less dense rock. Limestone with lots of internal cavities will curve heavily