Why a shotgun?
The fighting shotgun is the best match to the usual scenarios that fall under the heading of justifiable self defense shootings. You need heavy firepower at close range and the best hit probability you can get.
Almost all self defense shooting is at short range, and the shotgun, loaded with multi-projectile shells, is simply the best short range weapon. Its hit probability is twice that of a military rifle and nearly half again better than you get from a submachine gun. That is what was reported out of our military's JSSAP efforts and I see no reason to doubt it. My informal range experiments show the shotgun is fast to address close targets because of the confidence factor. The shotgun's margin for error allows you to shoot quickly.
As an expedient for longer range firing, the rifled slug, from a smoothbore shotgun, is effective at 75 yards, if the shotgun is equipped with rifle sights. When zeroed at 75 yards the typical slug's rise above the sight line is less than two inches, which is certainly manageable. The trajectory is 20 inches low at 175 or thereabouts, depending on the brand and style of the slug. Accuracy at such a long distances is, in any case, hit-or-miss, literally. At much beyond 100 yards, slugs are decidedly an iffy proposition. But they add versatility to the shotgun by turning it into a smoothbore musket that throws a large caliber projectile.
Some people, of the sort who like to devise training programs, have come up with the 'select slug' drill, which works like this: You look at the distance to the target and if it is farther than is ideal for shooting with buckshot, meaning the shot will spread enough that some pellets may miss the assailant, you insert and chamber a slug and aim carefully. My thought on the matter is this is a solution in search of a problem. In any likely scenario, you will not have time to select a slug and there will be no need to do so, because the distance will be short. Carry some slugs if you wish, but the likelihood of needing them is slight.
We know that most self defense shootings are at less than 15 yards, often much less. If you have a justified self defense shooting situation the distance will nearly always be such that the rigmarole of estimating the range and deciding which shell to use (and taking up precious fractions of seconds to do so) is pointless in the first place. As I have indicated elsewhere, the usual problem isn't shot that spreads out too much, but that most guns and shells pattern too tightly at the very close ranges that are usual in self defense.
The Body Armor Problem
Nothing you can load into a shotgun will penetrate body armor reliably. But since the armor doesn't cover everything, you are not helpless. The targets ordinarily available on an armored assailant are hands and arms, neck and face, legs and pelvis. Smallish shot with good pattern density, such as #4 Buck, will help you hit what the armor does not cover. That will work at short range; I don't know of a long range solution, except to suggest you figure out how to hide from the bad guys.
Most criminals do not wear armor, but enough do that you need a plan. Criminals who have been in the ground force military are well briefed on the stuff, and might even have brought some home with them.
It is perhaps counter-intuitive that smaller shot is better than the mighty double-aught, if the opponent is armored, but it is true when you think it through. The ability to hit a certain structure of the body, that is, to assure a hit within a certain smallish area, depends on pattern density, and that means small shot. Ideally you use the smallest shot that will give adequate penetration, a rule of thumb well proven in sporting uses of the shotgun.
Will you need your self defense weapon at 15 yards' range? Perhaps, and very likely not that far, if you need it at all. 30 yards? Not likely, but possible. At such distances the shotgun is the best small arm of all. 60 yards? Unlikely. 120? Very, very unlikely. Yet the shotgun, properly loaded and managed, can deal with all those distances. The problem with it is it is not very satisfactory for dealing with someone who is wearing a ballistic vest, beyond 20 or 30 yards. That is not a big problem because most hostile encounters, outside the battlefield setting, are at shorter ranges than that, and most criminals are unarmored.
It seems, to me at least, that it is sensible to think in terms of likely scenarios, instead of an unlikely worst case. If your armed encounter follows any likely scenario, the shotgun is the best thing you can have going for you.