I, personally, like the shotgun for HD. It's simple, versatile, reliable, and relatively inexpensive for practice. I've been hunting with one for >30 years, so I'm well familiar with the controls, as well, and have had many opportunities to fire at moving targets much smaller than adult human males. I'm not a "crack shot," but moderately confident in my ability to hit COM at HD ranges. I recently competed in my 1st 3-gun match, and there was no question as to my abilities: I'm simply much more proficient with a shotgun than anything else I shot.
Given my living conditions, a small apartment with neighbors, shooting through walls is a serious concern. Accordingly, I attempt to use HD loads that minimize that risk. Even if I lived in a suburban area, though, I'd still go with a shotgun. The possibility of shooting neighbors should be reduced as compared to an apartment, particularly if there are fences and the like in between houses.
I will concede that a shotgun is not as maneuverable as a handgun. I know that, in theory, untrained folks such as myself shouldn't go "clearing the house" before police arrive. In the event of a home invasion, my plan is to hunker down to the extent possible. However, I could not, in good conscience, leave my daughter unprotected at the other end of our home.
As for storing and securing a weapon, having small children in the house makes a big difference in the equation. I'm not foolish enough to believe that my daugher "can't chamber a round" or "isn't curious about guns." While both statements could be true now, I cannot guarantee that they will always be true. One of the things that I have done to avoid that kind of potential disaster is to teach my daughter about guns, and to let her handle them pretty much every time she asks. Guns are no mystery at our house. She knows that I carry a gun. She goes to gun shows with me. She gets to handle them when I'm cleaning. When she gets bored, she goes and does something else. That way there's no "forbidden fruit" aspect to them. I would encourage the OP to look at various storage & securing options in light of the fact that children are in the home.