An AR-15 is simply the best home defense weapon there is (along with tons of other semi-auto rifles). Lots of ammunition, quality rifle round, accurate. It greatly enhances your ability to defend your home and family.
The bolt or pump action shotgun would be FAR out shadowed by a capable semi-auto rifle with lots of rounds. That is a fact.
From a home and life defense perspective it is accepted fact that semi-auto rifle (with plenty of ammo) > shotgun > pistol.
This is a fact widely agreed upon by self defense experts. A semi-auto rifle in an intermediate caliber is easier to aim and will stop a threat more reliably than a pistol, and is more accurate and precise with much less recoil than a shotgun. Remember, you are responsible for every one of those shotgun pellets. The semi-auto, if loaded with soft point ammunition, it is also less likely to penetrate walls, which is especially important if you live with others or have neighbors close by. The ability to aim and fire successive shots quickly becomes important if the threat is shooting back, doubly so if there are multiple threats.
There are other benefits to them as well; people of smaller stature have a great deal of trouble finding long guns that fit them properly, and the adjustable stocks frequently found on AR-15s eliminate that problem. The rails found on most "assault weapons" make attaching a light as simple as possible, which encourages proper target identification, thereby reducing the likelihood of accidentally shooting someone you know. The semi-automatic action minimizes user error; there is no fumbling for the bolt in the dark or short stroking because you're panicked. You can also operate them with one hand should you become injured, have to assist children or move somebody to safety, or just need to open a door.
Having a 30 round magazine effectively means you will not have to reload; the 4-6 rounds held by most shotguns and revolvers and 10 round magazines the ban would require don't guarantee that. The police hit ratio is on average between 20-30%. These are people professionally trained to respond to high stress life and death situations with firearms. Using the low end of the police hit ratio to represent the average gun owner (I'd wager most gun enthusiasts would do better, but most gun owners don't practice that frequently so maybe they'd even be worse?), this is how'd they do with the following weapons:
Pump Action Shotgun - 4 Round Capacity (the basic Remington 870, one of the most popular shotguns, comes standard with a 4 round magazine): 20% of 4 rounds is .8 hits. This means that on average, one would empty the entire magazine and not hit the target.
Revolver - 6 Round Capacity: 1.2 hits before reloading. Lets keep in mind that a hit is not guaranteed to stop the threat, and multiple assailants is a possibility.
Ban Compliant Semi-Automatic - 10 Round Capacity: 2.0 hits. This is much better than the previous options, but that's still not a reassuring number. Frequently in fatal police shootings, the criminal is shot a seemingly excessive number of times. This is because police are trained to shoot until the person is no longer a threat. In a study conducted in Portland in the early 90's, they found the average person fatally shot by police was hit 9.3 times. This means they hit them twice, and still deemed that person a threat. Maybe all 9 weren't necessary in all cases, but that should raise enough doubt that 2 hits and thus 10 round magazines will always be enough for a law abiding citizen to defend his or her self.
Most of the advantages to "assault weapons" are heavily weighted in favor of self defense, compared to mass shooting. The mass shooter is prepared for the situation he (so far they've all been a he) creates; he has planned for it and isn't fearing for his life. He likely isn't going to experience the same overwhelming panic, adrenalin dump, and loss of fine motor functions attack victims experience. Thus, manipulating the weapon, whether it be for reloading or operating a bolt or pump, is going to be a minor inconvenience for him whereas it could be disastrous in a self defense situation.
His targets are not aggressively advancing on him, they are running away, trapped, or cowering. There is no reason to rush, so rapid fire is not a necessity. He is in complete control of the situation; he can take time to aim, reload, or even pick his nose if he wants to. The VT shooter reloaded over a dozen times. Until police arrive, they have all the time in the world. This most recent shooting lasted about 10 minutes; in that time 26 people were killed. That is one death every thirty seconds. There is not a firearm made in the last 100 years that isn't capable of doing that in a room full of defenseless children. The only limiting factors when you have that much time is your skill and how much ammunition you can carry.