First, various explosive devices have been used with slings in the past, and they are still used by paramilitaries. The typical example is the Winter War, where Finns used slings to throw Molotov Cocktails at enemy armour. There are also records of them being used in the Spanish Civil War.
A trained slinger can shoot a projectile with great accuracy and range, and while the rate of fire is slower than with a bow, it isn't necessarily slower than a thrower. So in a scenario where you want to launch a grenade at great range, they would work great.
The thing is, there are already better options for that in most cases. We have grenade launchers, rifle grenades, and the doctrines that make you close on your enemy very quickly. There simply aren't too many scenarios where lobbing a grenade far would be very useful. The main exception was where slings were actually used (until better weapons appeared and got reliable) - like the anti-armour use I already mentioned.
Now, add the training. Most people already have plenty of experience throwing things. If they can pass a ball, they're pretty close to being able to throw a grenade safely. Most people don't have sling training; and while training to use a sling isn't quite as hard as training to use a war bow, it isn't trivial either. Rifle grenades are a lot easier to use.
Slings do require some clearance. It's not impossible to use them indoors, but it's definitely awkward. And imagine how a bunker would have to be built to allow you to sling grenades on the enemy (they did have grenade chutes). You also have a bit less control over the trajectory - while you can target them precisely, you might have trouble with enemy cover, even trees.
You can't just use a hand grenade in a sling - that would be quite dangerous. You need a shot designed to be fired from a sling, and that means yet another piece of ammunition that gets through your whole logistic chain, that has weight you have to carry, and there are pretty much no circumstances under which it would be better than a rifle grenade or a hand grenade. Now, it might be possible to design a grenade that can be used both with a sling and as a hand grenade, but it's another piece of complexity for an explosive - hand grenades are plenty dangerous already, and they need to be very cheap to be useful. Mind you, neither would be a problem in most modern armies - but most modern armies (again) have better weapons.
Finally, you need to ask how grenades are actually used. A typical anti-personnel granade has two main use cases - getting enemies out of cover, and killing clusters of soldiers in closed spaces. For both, you already need to be pretty close, and in a cover of your own. Why use a sling to throw a grenade when you get artillery ready to rain accurate fire on the enemy? Are you going to shoot a grenade through a window? That's the perfect use case for a rifle grenade :)
With the grenadiers of old, they were used against enemy formations, mostly in the clear with no cover. The grenades had rather long fuses, and the enemy had limited maneuvering to get away from the grenades. And you had two hundred grenades being thrown at you at once - quite a rain of explosion. Of course, slings aren't easy to use in a tight formation, and armies relied on tight formations for defense against cavalry, so this wasn't really used much in a real battle anyway.