actually, sir, in my opinion a home with kids is the perfect place for a loaded gun. if you live alone, you are free to choose what to do with your life, a choice you dont have with other's life, life that is your duty to defend.
and an unloaded gun is not going to be any good in case of home invasion
lets imagine you have a swimming pool in your garden. what would you do with the kids, prohibit them from jumping in, or teach them to swim? a neighbour of mine took the first approach and lost one of his 4 kids...
now, back to the original question, i wouldnt consider a shotgun for home defense. too loud, too long and clumsy, possibly not enough capacity. my choice would always be a high capacity handgun, maybe with a torchlight, maybe with a red dot. probably heavy ball subsonic ammo to reduce noise.
regarding overpenetration, anything that is going to penetrate an intruder will penetrate doors and plasterboard. so there must be a good situation and tactical awareness, and never shoot in the direction where people living in the home might be affected. a good tactical plan is a good idea.
home invasions are a nasty business. i had people breaking into my cellar, which is not connected to the rest of the house.
i was home alone, i have an alarm system (that i built and programmed), they probably heard me getting up and left before i could see them. or perhaps the reloading equipment in my cellar, and thousands of cases ready for reloading, sent them a subliminal message.
i used to leave the doors open. not any longer.
home invasions in my area used to be exceedingly rare. its still quite ok, but things are changing.
https://en.wikipedia.org/wiki/Rupperswil_murder_caseI woudn't keep a loaded gun in a home with kids.