Whatever you put in a shotgun and whatever gauge you use will work most of the time because most encounters with bad guys are ended by the prospective victim just having a gun and showing readiness to use it. Additional cases are ended by the firing of the gun resulting in resulting in no hits and the bad guys running away or some type of wound, resulting in the bad guys running away or surrendering. In all these cases, a .410 with 7&1/2 birdshot will work.
The problem arises when someone faces an assailant or assailants, who are loaded on drugs or just especially determined and the gun has to physically stop them. I don't get the whole birdshot for home defense thing. The studies are out there, easily found on the internet, where birdshot has been tested on ballistic gelatin and water jugs and other materials. Heavy clothing severely limits birdshot"s effectiveness, leather jackets especially. Birdshot gives a very nasty surface wound and very limited penetration even at close ranges. This is a fact as demonstrated by the above referenced tests.
The reason birdshot doesn't "over penetrate" is the same reason that it is so limited in it's ability to stop a real threat It doesn't have sufficient penetration to do the job.
For those who wish to beat their chest and declaim how deadly their birdshot loads are, what ever works for you is your business, and the odds are you will never have to back up those statements. There is a big difference between what people choose to believe and what can be demonstrated.
Probably won't even need to fire it, anyhow.