It seems using a sling is far more effective than throwing things with your bare hands. As far as I know, slingers were widespread in the Ancient era, as well as in the Middle ages:
However, in XIX-XX hand grenades were thrown, well, by hand:
Googling "sling grenadier" gives a bunch of fantasy troops. The only one mention was the Spanish Civil War sling grenadiers:
Slings were way more compact in comparison with a grenade launcher. However, using slings for throwing grenades wasn't so popular. Or was it? And why?
"There are situations where you actually can't use a sling (trenches, indoor, under fire, etc.)" - I'm not talking about using a sling every time INSTEAD of throwing a grenade by hand, but in addition to. It's like a rock - you can use it as a sling ammo, but you can still throw it by hand, no problems.
Slings have low accuracy - A sling target used for competitions is ~1.5 meter wide. You are supposed to hit it from a 90 meter distance. So, slings have decent accuracy.
Grenades have another use-cases - indeed, you don't need a long distance to, say, throw a grenade into a room. But there are still use-cases where you need that distance. That's why we have grenade launchers. But you can't carry a grenade launcher in your pocket. Also, a sling are way cheaper.
Fusing problems - I know grenades have safety handles and you're supposed to grip it when you throw a grenade. Using a sling will make this mechanism unsafe and doubtful. But that is kind of deductive fallacy - grenades have safety handles BECAUSE we throw them by hand. If slings were used, there would be different safety mechanisms.