Use buckshot for home defense. It is already designed to limit pentration. Birdshot may well work, but it may not and there is no reason to not choose buckshot. There is absolutely no downside. Even buckshot fails far more often than many want to admit.
Under perfect conditions any load from a shotgun is going to be a serious threat, but once stuff starts hitting the fan you cannot count on perfect conditions. Plan for the worst and hope for the best. Planning on #6 shot working is really hoping for the best.
An intruder wearing heavy winter clothing, or a leather jacket could easliy soak up a load of birdshot at fairly close range and only get a few pellets to penerate the skin. None to any vitals.