##Other alternatives are better
We can see that many grenades were designed to be thrown by hand. Sure, attempts to throw those using a sling would end badly. However, that does not answer the question why thrown-by-hand design was adopted in the first place.
Let's compare sling-thrown grenade to alternatives. I suppose grenade with a rope attached would be better than sling-thrown in almost any way. Mechanics are the same, lesser risk of grenade slipping out and so on. U.S. Army Improvised Munitions Handbook has a section about doing that (adapting hand grenade to be used with a rope).
What else is similar to grenade-on-a-rope? Grenade-on-a-stick of course. Grenade on a stick requires even less skill to launch and still provides increased range. So, if you think about designing grenades for a sling and equipping your troops with those, you're better off giving grenades with ropes attached, and grenades on a stick are even better than that. And those have actually seen their uses!
Sling grenades could probably provide more range than stick grenades. However, you have rifle grenades for that which require less skill to use. They also require less room, which is important because:
##You don't have much room in trenches
There were threnches, lots of those and you should generally be in one of those too. You can throw a regular grenade while you're in a trench, albeit less effectively. Throwing grenade on a stick would be harder (because there's not much room). Additionally, accidently chipping trench's wall with tip of a grenade is easy and would set off some early ones immediately.
So, there are 2 and a half options - "regular" grenade, stick grenade and sometimes rifle grenade. Sling grenades would be inferior to those, especially in trenches which were prominent when grenades were being developed.