I've never found the Slingshot useful in any game that isn't Ocarina of Time. In Ocarina of Time it has a use as you are unable to use the Bow as Young Link, which I thought was fair. It let you kill Skullwalltulas and other Skulltulas with ease as you didn't have to get close to hurt them, and since they appeared in a few areas that you needed to visit as Young Link, it was useful to have with you.
In Twilight Princess, it's completely different. Once you have collected the Bow the Slingshot becomes a useless item that has no worth, although there are a few times you do have to use it before then, mainly it has a similar use as the Slingshot as Young Link in Ocarina of Time, except it's not used anyway near as much. When you do get the Bow, it's just an upgrade really since it has greater power, a much larger range and can be equipped with several other items such as Bombs and Hawkeye.
Bow runs out of arrows, so natrually the slingshot would make a decent replacement it may not be powerful but it's still a usefull projectile item when the bow is not able to be used for whatever reason, there by making it the exact opposite of useless.
The Bow has a much different use than the Slingshot, whilst they are both projectile weapons, you generally need the Bow to cause damage to the enemy you are using it against, not to stun them. Even if you are against an enemy where it would be useful to stun them, the Clawshot does a good enough job to make it worth while, it has infinite uses and can be used at roughly the same rate as the Slingshot. So really, the thing you just stated that makes the Slingshot useful is only really there for the period of time between the mini-boss in the Goron Mines and the Lakebed Temple's mini-boss, half of which you are trapped in Link's Wolf form. That and the fact that arrows are plentiful in Twilight Princess in most areas where there are enemies.
A few others have said that it's been created so that it will be more useful at close range whilst the Bow is more useful from a distance, but I'm not sure how useful I'll ever really find this. I've never pulled my Bow out in the middle of a fight to use it before, so I don't see why I'd start doing it in Skyward Sword. I'll just have to hope that it does happen to be a use otherwise it's going to be another attempt at the Slingshot that has failed.
