People aren't sticks of wood, for starters.
Nobody said birdshot wouldn't do damage to an intruder, and nobody said it wouldn't make a person decide they don't want any part of remaining in your house, and nobody said it can't kill, especially at close ranges
HOWEVER -
An inventive lawyer of that intruder that you shoot and don't kill with birdshot inside your home may very well decide to attempt to cause you a great deal of financial trouble for something like purposefully injuring, maiming or doing great bodily harm LESS THAN DEATH on his client - Intentionally damaging someone to the degree that they will have permanent injury or disability, or a long, slow, painful death is cruel regardless of whether he's breaking into your home, or that's what the lawyers will try to make the jury believe. Many of them probably will believe that - Most people have a hard time where violence, death and other "bad juju" is involved, and while most everyone here would probably vote "not guilty" in a home invasion scenario, you don't get to pick the jury - the lawyers on BOTH sides do.
You wouldn't use birdshot on a hog, or a deer would you? No - you'd go to jail for awhile for violating game laws, lose your right to hunt or own guns, and likely lose your job, home, and everything else you hold dear - Let's face it, when you're fighting for your LIFE against an intruder intent on taking all those things away from you and more, do you REALLY want to use birdshot that will only penetrate an inch or three into the bad guy?
Sure the wound will be big and nasty, you might even take half his face off and destroy his eyesight, ability to taste and smell, even his hearing - but head shots on a rapidly moving target are difficult, so you're probably going to try for center mass - not one of those itty bitty tiny pellets has enough mass to pass through the breast bone, much less the entire body - it could definitely be argued that by using birdshot on an intruder, you weren't fighting for your life, you were intending to make the intruder pay for breaking into your home for the rest of his life. (I know I know - captain obvious)
My point here isn't that birdshot at close range from a 12ga shotgun can't be fatal or that it won't do the job sometimes - my point is that if you're going to do something when your LIFE is on the line, use what has been proven effective. Use what has the highest chance to stop the attack. If you're using a 12ga, use something designed for a 125lb or larger beast - because that's what you're facing - not a 3-12lb bird.