The idea is that the lead pellets that hit the bird but don't kill it eventually poison it later. Given that bird hunting is generally firing into a large flock it's fully expected that few pellets will hit birds and not take them down
Wrong and wrong.
If the bird dies later, it's a result of getting shot, not from having a bit of metal left in it.
As previously pointed out, lead left embedded does not result in lead poisoning - lead is fairly inert. A lady at my previous employer had a brother with a bunch of lead shot in his leg from an accident as a kid - in there for years with no problems.
Lead only becomes medically dangerous (other than the whole kinetic energy thing) when it oxidizes, and that happens in the stomach, and to a lesser extent in the bloodstream.
The reason for banning lead shot is that the shot that misses falls into the water, and bottom feeding ducks later ingest it (see stomach acid). This is the waterfowl poisoning route.
Shotguns do not produce some huge pattern until the kinetic energy is reduced to the point the pellets aren't dangerous. At the range you are likely to take down a waterfowl the pattern is still quite tight - a few inches. It isn't spray and pray like you see on TV.