If you have dynamite, then you either have, or are on the very cusp of discovering, gunpowder. So while, yes, you can sling dynamite (catapults and ballistae were used to lob grenades all the way up to World War I), the use-case for that capability is extremely short lived.
Your point about slings requiring skill is valid, but consider that: also like archery, slings were a common tool to obtain food, meaning that large portions of the population are versed in their use from a young age as a matter of practical tool training for civilian life. On top of this, skirmishing, which is what someone slinging explosives will be tasked with - the risk of friendly fire incidents is obscenely high otherwise - is a high-skill type of warfare already, so training/selecting for good sling/archery skills isn't as marginally costly as you might think.
The musket and it's brethren also only worked the way you describe when used in massed formations. An individual peasant musketeer couldn't hit the broadside of a barn except by luck - and that's only partially the fault of the musket. Proper sharpshooting took the gunpower boomstick and brought it straight into the same realm as archery and slings: a high-skill requirement to get the most out of the tool.
You can train slingers for a very short period of time and use them en masse to similar effect: quantity has a quality all its own.
Incendiaries are likely to be developed MUCH earlier than explosives, so a use case for "I want to set that person on fire, but I just can't reach" happens almost immediately after fire itself is discovered. Again, slings can do the job but the engineering requirements on the vessel are severe - it needs to resist the compression of the centrifugal force applied by the sling, and then break open on impact. On top of that, incendiaries (barring stuff like thermite or phosphorous), are best used in very large quantities: small fires are easy to put out and slow to spread. Large fires get out of control without any help at all.
If you've got powerful incendiaries like napalm, thermite, white phosphorous, or magnesium - which you might be able to discover/develop before explosives - now you're cooking with... liquid, I guess? As long as the vessel design can do what you need it to do, even something as simple as a strip of steel acting as a spring-arm (world's easiest catapult) will happily put that down range.
Long fuses that you hold onto to impart momentum to the bottle are likely better than slings because of the coordination required for proper sling use.