Biden jabs aside (I wouldn't worry about HD if I had a SS detail living with me either), a coach gun is still a shotgun, and as such pretty effective at most everything.
Also, as they don't have a long action stuck in the middle you get a shorter gun out of the same length barrel. For inside a house, shorter and more maneuverable is generally better.
The only downsides I see are limited capacity (generally 2), and the inability to keep them loaded safely.
From a defensive standpoint, I don't see either as a major problem.
Assuming you're hunkering down in a "safe room" of one type or another, it seems highly unlikely that you're going to need to fire more than 2 barrels of buckshot at anyone coming in. And if you don't wake up with enough time to pop a couple shells into a shotgun you're probably SOL anyway.
A huge advantage out in the sticks is that you can load the barrels separately. So, if you're going to check a noise outside you can have rubber buckshot in one barrel (for a bear) and lead shot in the other (for foxes/racoons).
Anyway, that's my take. YMMV
si vis pacem para bellum