It's hard to square this circle but it needs to be for shooting to continue. It goes something like this, viewed from the outside:
1. Millions of game birds are bred every year to be shot.
2. This is OK because they are eaten, there are biodiversity benefits, it makes money.
3. If 2 isn't true, then 1. isn't OK.
Lead shot potentially stops 2. being true, because shooters don't eat all those game birds. This is because of the scale of driven game breeding and shooting, and is true even if most shooters don't actively join in.
Basically, the "value chain" is all wrong... Shooting and eating a harvestable surplus is one thing. Creating excessive supply is another. The latter in the case of game birds undermines the former in for instance the case of deer. Yes it's simplistic and certainly on the ground there is lots of nuance, but perhaps not from the outside, which is where the argument needs to be won.