Mmm. Most typically, peltasts had javelins and not slings or bows.

I'd like to point out that peltasts didn't depend solely on their ranged weapons, but on their light weight equipment and loose formation. They were more free to move around the field than their heavier-armed and tighter-packed counterparts. As such they would typically move to short range and throw missiles at opposing hoplites and phalangites, disengaging if pursued. At the battle of Cunaxa, a body of peltasts allowed enemy cavalry to pass through their ranks, being able to move aside thanks to their agility, and throwing javelins into them as they passed.

I'm not familiar with Gaugamela, but from other battles in that period, the peltasts being used to screen Alexander's advance wouldn't have been a nice dense formation lobbing missiles at standoff range but rather a porous and chaotic mass. If you order horse to fight in such a way, their advantage of speed is negated. If Persians charged into the peltasts, they would have gone in like a knife through jello and then been harried by the agile light infantry hitting them from all sides.

Sort of losing the thread of the original point though, which was that slingers could be effectively trained to a high enough skill level to serve as soldiers in a useful amount of time. Certainly the shepherds, having ample time and motivation to practice the sling would be very good with it and your first choice if available... but I cannot believe that ONLY shepherds were chosen as slingers when there were thousands of such troops in an army of the day.