Well, it depends. If you shoot a target with a shotgun your chances of hitting your target are greater than if you shot the same target with a .22 cal. The point here is the amount of projectiles shot at the target in a given time frame.
Like you've mentioned, the key is the "time frame" and "the amount of projectiles shot".
Assuming a single shotgun slug holds 20 pellets inside, when a shotgun is fired the 20 shots are simulatenously spread out in a wide pattern which is evidently intentional. Shotgun pellets are meant to spread out, and by doing so, while covering a wide range there is no possibility that the entire 20 pieces from a single slug can be "grouped" at a certain spot. In order to compare the general accuracy of the .22 as opposed to a shotgun slug,
the experiment should fire not a single round and compare that with a 20-pellet-filled slug, but rather fire 20 individual .22 rounds and compare the accuracy of 20x .22 against a 20-pellet slug.
Therefore, when one compares a 20-pellet, single slug against a single .22, that's not comparing
"accuracy", it is comparing
"efficiency".
IMO, that is why the
"shotgun analogy" is wrong. People treat the dispersion pattern from an aircraft gun as a "shotgun" - firing multiple rounds simulataneously and intentionally in a dispersion pattern - when in reality, the pattern formed from an aircraft gun is a result of multiple rounds fired one at a time, over a certain given time frame.
Ofcourse, in reality the "shotgun effect" does exist, albeit only in planes with multiple armaments spread out at a certain distance - particularly across the wins. However, this "shotgun effect" is not a result of dispersion of an individual gun which lands at a "shotgun pattern", but rather an effect caused by multiple guns firing at the same time. In short the reason behind what produces the effect is entirely different from what we're discussing in this thread, and its basically misleading.
The accuracy of the individual pellet is abysmally low than compared to a .22 round which is aimed and fired. However, the efficiency of 20 pellets fired in an intentional pattern that spreads
simultaneously, is greater than a single .22 round fired, or .22 round fired 20 times. If we situate this analogy directly to an aircraft gun, the "shotgun effect" is a result of multiple guns firing simultaneously - we're not talking about that "shotgun" here. We're talking about a "shotgun pattern" produced by an individual gun due to its own dispersion (not, by an intentionally spread pattern from multiple guns).
Now, let's continue with the .22 vs shotgun example.
Unlike a shotgun pellet, 20x .22 rounds fired individually have a chance that all 20 rounds might connect, but it also has a chance that none of them might at all. The shotgun is different. It fires multitude of rounds simulateneously, and therefore while there is no chance for all 20 pellets from a single slug will connect at a target (ofcourse, assuming that the target is smaller than the magnitude of the pellets spread) , there is a significant chance that "some" of it will connect. This is why you can't compare a shotgun with the dispersion pattern of a gun. The dispersion pattern created by an aircraft gun represents the probability, not a result of a directly predicted pattern.
Therefore, when someone has aimed reasonably accurately, so that the target is set in the center of the dispersion (which means the target is aimed and placed in a position where the bullets have the highest probability to hit), in this case the dispersion is
malevolent to the probability of hit. This is because dispersed bullets that diverge away from the center of the dispersion pattern (which means "highest hit probability") will miss.
However, when someone has aimed wrong, so the target is set apart from the center of the dispersion pattern - only then does the dispersion increase the chance of hit. Because, the diverging round that strays randomly off from the center of the dispersion pattern contains a miniscule chance to connect with the wrongly-aimed target, since the target was out of aim in the first place - which otherwise, in a perfect vacuum world without dispersions factors such as air, vibration, wing twist, etc etc.. there would be no chance to hit at all.