The question about which type of weapon is best is certainly opinion based. It depends on circumstances. What is not so opinion based, and stays consistent throughout different situations, are some principles: the importance of capability and familiarity with the equipment. Whichever weapon is more capable in the situation and whichever you are more familiar with is the way to go.
A capable weapon is one suitable for the conditions. For the first short chunk of time, perhaps the firearm is the way to go, but in the years ahead without infrastructure for maintenance, reloading, and repair, something like a sling would be of greater value. 'Necessity is the mother of invention' and in this case, you'll find that the tools different cultures made most use of were the ones which were most capable in meeting the demands at hand. If you're looking for a well rounded tool that is most widely capable to meet the dynamic and long-term challenges of long-term survival, you'll probably want something like a knife or axe, and after that, whichever best suits your environment, needs, and abilities (which is where 'firearm vs. sling' becomes moot - depends on many factors about you, probabilities about your surroundings, probabilities of near and distant future scenarios).
Secondly, your familiarity is important in deciding which is more valuable. If you don't know how to operate a weapon, it can be both useless and even dangerous to your own well being (consider if you become disarmed by someone more familiar with the spear you were just holding, or if you sling a rock in the wrong direction). If you don't know how to maintain, reload, and repair a weapon, how long will it serve you? All that said, familiarity can be picked up, especially if the stakes are high. So then we need to consider, what's the learning curve like in terms of becoming familiar with the different tools, especially considering the lack of infrastructure? Good luck figuring out how to repair a flint lock with no infrastructure if you don't already know how it works - not to mention actually doing the repair.
With all that said, if you had to pick one, a decent rule of thumb is that more primitive tools (slings, knives, spears, axes, bows, darts) have been tested further by time and can be more reliable as a place to start for the long run, but in the short-term the more advanced tool can handle more advanced challenges in conditions it is capable in. Do note that humanity evolved to use more advanced tools as more advanced demands called for them; realistically you will never 'just pick one' and depend on it for survival for the remainder of your time. Challenges change, conditions change over time, and so (hopefully) will you. In more common, non-apocolyptic situations, many experts emphasize how your survival in wilderness depends more on your craftiness, mind-set, and adaptability than on the tools you have.