From purely a terminal ballistic point of view, the shotgun has tremendous potential as a fight stopper. At a typical household maximum range of 20-25 feet, the pattern from an open choke gun will typically be 7 or 8 inches, and you can expect the action to be closer than that.
Anyone who thinks that birdshot isn't a fight stopper should reevaluate their position; you're putting an ounce to an ounce and a half projectile on your target. Compared to a pistol round, and many carbine rounds, that's significant. If your first shot doesn't produce the desired effect, the math is pretty simple. Shoot the bad guy in the pelvis, you've anchored him; if he doesn't give up at that point, you can at least move your family to a safe location. Shoot him in the chest or the head, and his participation of the fight is concluded.
That being said, a variety of shells in a sidesaddle might prove beneficial. If a family member is being held by the badguys, a slug provides the precision necessary for a clean hit without the risk of injury to the hostage from a stray pellet that's outside the normal pattern. Any gunfire within a dwelling is potentially dangerous, so consideration must be given to the shot angle and direction to avoid the tragic consequences resulting from your gunfire injuring an innocent person. The badguys are under no such constraint, so if you're unable get your family to safety and shooting can't be avoided, its best to end it quickly.
A protracted gunfight benefits no one; shoot first, shoot quickly, and shoot well. The importance of fire discipline cannot be overemphasized. You don't shoot more than you need to, and if you must shoot, you do so with the direction and shot angle in mind. Its important to think about this stuff before you're actually faced with it, and if these threads have any value at all, its not the argument between buckshot and birdshot, or the argument between shotgun and carbine, that's of value. The value is that you might consider the possibility of a home invasion and come up with a plan.
If you know how you're going to respond to a particular threat, if you know what you can do with your gun, and if you already know where the danger areas lie within your home, you're well ahead of the curve.