It's really a misconception that they (shotguns) don't
+1. At social distances the pattern of a cylinder bore shotgun is only a few inches in diameter. Only when you've got a decent round count over a lot of different training sessions will you be able to "point" and get good accuracy cold. Even then, you're really actually aiming, except your aiming device is muscle memory and your training instead of the rib on the barrel.
Avoid slow and heavy projectiles - or fast heavy ones. You'd be surprised how much wall a pistol round will punch without destabilizing. Either will go through tons of dry wall readily. Go with a .223 in 40gr - part of the reason M4 style ARs experienced massive adoption rates in the last decade with SWAT teams is that the projectiles tend to break up when hitting a barrier. In a good, mid length gas system M4 clone.
A 12 gauge loaded with some high brass turkey shot would do the trick. As for people recommending the 20 gauge, only if you're planning on the women or kids using it. Any grown man should be able to manhandle a 12 gauge unless you're shooting 3.5inch shells.
In general, long guns in a carbine length would also be best. It is not just longer sight radius that makes them more accurate than a handgun, they point a lot more naturally and rapidly due to multiple points of contact (chest/shoulder, strong hand, weak hand). In other words, even without using the sights you'll be more accurate. Your first shot will be more accurate and your follow up shots will be more accurate with less split time.
I hope there's someone else in the thread who has done training in dynamic entry, battle drill 6 or reflexive fire who will back me up on long weapons. I can't emphasize enough how much better a long weapon is in every way whenever you can have one.