Hi, aujrb --
I divide home defense firepower into "infantry" when you have to go mobile, and "artillery" when you ensconce in one position. The handgun performs the "Infantry" function when you have to move, and the HD carbine or shotgun works best when you need maximum power and will be firing from a static position into a known quadrant of fire, like "artillery."
If you have to search, gather the family members from their bedrooms into a "safe room," or simply get up to answer a late-night knock at the door, I recommend the handgun. It leaves at least one hand free to operate cell or hard line phone, flashlight, light switches or doorknobs, carry the baby, or put one hand on the shoulder of Great-Grampa with Alzheimer's to guide him down the hall to a safer position.
Once the family has gathered in the safe room, if gunfire is needed you can deliver it with more power, more speed, and more accuracy from the SG or carbine. In this situation, you are less likely to worry about being jumped and having to struggle for the gun with an intruder who was waiting in a closet to pounce on you than you would if you were moving or searching through the house. A long gun is more difficult to retain in a struggle, particularly if you only have one hand on the gun (remember the flashlight, the phone, carrying the baby from nursery to safe room, etc.).
Another thing to remember is that answering the door with an 870 or AR15 is going to turn ugly if it's the cops who rang the doorbell, and good people will get the wrong impression if they see you step out of the house to look around while holding a long gun. In either situation, your handgun can be discreetly concealed but immediately at hand.
Real world home defense is a lot more complicated than "grab gun and start shooting," and that's why I suggest different tools for different tasks within the strategy framework of home defense: handgun as "infantry" when moving about, long gun as "artillery" when ensconced.
Hoping that clears up any confusion,