Arguments about how slinging is outdated and wasn't done on a large scale since medieval times are inherently flawed. We still use knives, you know how old those are? Don't even get me started on how old throwing is.
A lot of people are trying to find a way of explaining why a sling wouldn't be better than a high-low grenade launcher system or throwing a grenade by hand. Perhaps because if it wasn't, then why don't we do it? But the military ordinance departments don't always make decisions based on what is the best for combat, but what is best for the doctrine/budget.
Slings are difficult to use. It seems one of the best examples for sling grenadiers was from the Spanish civil war, I would bet most if not all of those guys came from the Balearic Islands in Spain. The Balearic Isles have been known for producing talented slingers for thousands of years and they keep that torch burning to this day, slinging is a popular past time there. So most Balearic men already are skilled and confident slingers. But that would have taken them years to get that good. You can't really put a volley sight on a sling, so judging where the grenade will land has to be done with the naked eye. It's a bit like hip firing a gun without looking down the sight, except harder because you also need to worry about getting the timing right.
It takes a lot of practice to learn how to sling. militaries are not willing to spend that much time and money training Joe how to sling a grenade when they could just give him an 8-pound grenade launcher for the same effect. Yeah, just having to carry around a light piece of string sounds great, but the brass doesn't really care about your back pain.
Now, are slings actually worse than a grenade launcher or hand-thrown? A quick google search says that a sling has a range of 360-400m and a grenade launcher has an effective range of 350-400m, so about the same. As for issues about space and safety, watch this video https://youtu.be/yJ3bBkRIJNU. In particular, the overhanded technique and the two-handed technique look like they would be most applicable. They are fast, don't take up space next to you, and if the grenade is dropped accidentally it will only end up somewhere in front of the slinger and never behind them. They could potentially be done while the sligner is half obscured by the cover as well. Another benefit is versatility. 40mm grenades cannot be used as a hand grenade in a pinch, nor can most rifle grenades. However presumably a sling grenade could. Then troops would only have to carry one type of grenade that can be either used by hand at short distances or with a sling at long distances. Another obvious advantage is that slings are very quiet whereas grenade launchers are very loud. One of the major reasons why the US military switched to using 40mm grenade launchers instead of bazookas was because they were more discrete due to not having a backblast or a rocket trail leading back to the shooter. A sling doesn't even have muzzle flare. Also unlike a grenade launcher, a sling can be used to throw basically anything that would fit in the cup. Rocks, improvised grenades, hand grenades captured from the enemy, spy camera drones, etc.
Honestly, I do think that slings are a bit underused by militaries today, perhaps they just don't think of it when writing doctrine and tactics. While not every soldier can be expected to learn how to sling, imagine if special forces were also skilled slingers. What if every single member of a company also had a grenade launcher that only weighs a few grams and it was completely silent? Any lightly equipped squad could lay down a barrage on any target within 400m and the target would have no idea where it was coming from.