Both have their advantages and disadvantages over the other.
I have a similar mindset, we have 20G mossberg pumps strategically stashed around the house for home defense, of course we have handguns and AR's pretty readily available as well.
first choice is always the pump shotgun..
My load out is #8 bird first, then #00 buckshot.
theory behind it is birdshot will only penetrate so many layers of drywall, so downrange risk of family in a 'gotta shoot' moment is minimized, pepper someone's torso with birdshot while you maneuver to a 'safer' downrange backdrop, or wait for another family member to provide backup.
I've been peppered with #8 at around 100 yds and with several layers on in a bird field. It'll make you say words you don't want grandma to know you know, even at distance.
It shreds yotes and possums up to about 20 YD, so I know what damage to expect.
Still don't want to use one in my house, but ain't gonna risk my wife, kids or grandkids' lives on a whacked out crackhead..