There's alot of data that indicates that a 5.56x45 carbine is a better choice for most HD and LE situations than a shotgun, at least if the operator is up to the task.
The 5.56x45 is less likely to overpenetrate than buckshot (or pistol bullets). All will go through home walls, but the 5.56 bullet will be in tougher shape after going through the first wall and is less likely to continue through more walls. So, I wouldn't chose a shotgun (or pistol) based on penetration.
As a note on that point, IMO the idea of using birdshot, Glaser slugs, etc., is very unsound. While there may be limited cases where they will work and might even be optimal, there is no guarantee that those conditions will exist, or that they will not change, and leave you inadequately prepared. I want my PDW loaded with projectiles that will penetrate with a margin for error. YMMV, but if the bullet hits the attacker sideways and first penetrates bone and muscle in the arm, for example, a lighly-penetrating projectile may not halt the attack.
The idea that you don't have to aim a shotgun or use the sights is no longer considered good training doctrine (partly because of what happens if you miss or overpenetrate). And, many HD/LE shotguns now pattern more tightly: for example, the VangComp choke, and Remington's move to IC choke (at least for the Police Magnum models). So, although it's still true that there is a greater margin of allowable error with a shotgun, IMO it's not a huge factor.
Certainly, for user-friendliness, the shotgun has no advantage. Its recoil is significantly heavier than a 5.56 carbine, and the shotgun is probably a little longer, too, even with an 18" barrel. The shotgun isn't going to be any quieter (at least not to notice) indoors.
There are some advantages to the shotgun. Domestic models are (mostly) less expensive than even a Mini-14 and much less expensive than a quality AR-15. And, a pre-ban AR-15 may soon be irreplaceable -- worth thinking about, since the defense weapon will almost certainly spend a good deal of time with the local authorities and may never be returned.
Shotguns are probably slightly more sympathetic (depending on how configured) with the jury than an evil black assault weapon.
Shotguns probably slightly (you can flame me on this point, but who really knows?) more aurally and visually intimidating to your attacker. Personally, I find _any_ loaded weapon pointed at me very intimidating...
Shotguns are more flexible than either handguns or rifles, because you can switch ammo types readily (buck to slug, for example). However, using slugs assumes you've got the training both to execute the change and to deploy the slug accurately, which +99% of home defenders don't.
My suggestion would be that a post-ban Mini-14 would be the best pick (but better make sure the mag(s) is/are reliable feeders). I wouldn't go for full-power hi-power rifle cartridges like .30-06 or 7.62x51 (too much penetration). An SKS or AK-type rifle might be a good choice. In fact, a Winchester or Marlin lever action .30-30 has similar ballistics and is cheaper, easier to replace, easier to play the sympathy card with. Not sure how the AK or lever will penetrate compared to 5.56, but I'll bet it will be significantly greater.