Don't have a physics degree here, but I am pretty sure that once whatever is making the projectile (ammo or baseball) departs what is thrusting it forward (band or pitchers arm) it instantly starts slowing down, with nothing pushing or pulling it forward it can't go faster (air resistance and gravity).
I'm actually a baseball fan and a catcher on an adult wood bat league up here in NY. Pitchers can do amazing things using the laces of a ball and air resistance, but they simply cannot do anything to make the ball go faster once it leaves their hand. They can put spin on it to make it drop, curve, or cut, but not go faster. They can do one thing which might make it "appear" to go faster, which is if you throw a 4 seem fastball, it will keep the ball on the same plane vertically longer which might make it appear faster, but it isn't really faster, it just stays UP longer.
Baseball physics can be funny. Let's take a hitter instead of a pitcher. Should a hitter hit a 50oz bat that he can only swing 20mph, or should he use a wiffle ball bat which he cans swing 100mph but when it contacts the ball it send the bat backwards instead of the ball forwards. The variables are great but in theory a batter wants to use the heaviest bat he can swing with a high velocity. Then once he has that, if he wants the ball to go further he wants to hit the ball in a manner which puts back spin on it to keep it in the air long enough to get out of the park. Think tennis, top spin allows them to hit the ball fast and then it dives down to stay in the other end of the court, If the tennis player puts back spin on it, it tends to go slower, but it can often goes long and out of bounds because it slowly floats a long time and stays in the air longer.
Of course I imagine most of these dynamics are taken away in slingshots as most of the ammo is smooth. I can tell you clay balls definitely curve, but we didn't make them perfectly round or perfectly smooth. But if normal ammo is perfectly smooth and round and the release from the pouch puts no rotation on the shot then the balls speed and distance is totally dependent on its exit speed, weight, air flow and gravity. However it should go relatively straight (side winds)! Unlike a baseball that has laces so the wind moves it all over the place (aka knuckle ball).
But I still believe that the point you are looking for, where it is no longer accelerating, nor slowing is the exact point where the bands stop 'pulling' the ammo forward, which in most cases is probably actually the length of where your bands connect to the fork and the pouch. I think if we could slow things down enough we could see that at that "slack"' point, the ammo disengages from the pouch and the pouch is just following the ammo through at that point. I could be wrong, and if so, would definitely like to understand better.
O, and if you ask why the ammo disengages then, it would because the force propelling both the pouch and the ammo forward stops at that point and the ball has less air resistance than the pouch, so the pouch begins to slow fast.
Hope this helps, hope it makes sense, hope it is correct, but if it isn't, I don't mind learning something new! Or thinking a different way.