shooting with a bow , it's in the body with a large broadhead and it depends on blood loss and breaking back and other bones.
with a gun , it depends on lots of pellets in a small target.depending on breaking the neck of hitting the brain.
all that being said, since you ask, you are not of the opinion that the 410 will do the job either.I think a 410 just doesn't have the # of pellets and energey to do the job.I shoot quail and squirrels with 410 3" #6 and some times loose one because of the above said problem.all the shots I take are real close on the squirrels and some take the second shot. most times the second shot is not there on a big old gobbler.
Since I asked what? I didn't ask anything...and I know how an arrow and birdshot kill and there is no way anybody will ever convince me it's more difficult to kill a turkey with a .410 and birdshot that it is with a bow and arrow. I've also hunted small game with .410 and 12ga...and have had game run off with both..just as everyone else has that's hunted very much with shotguns. That is why I didn't recommend a beginning turkey hunter start out with .410..however...there's been probably who knows how many thousands ofgobblers killed with one over the yrs. If you use it like a .410 I see absolutely no problem. If you can't control the urge to shoot out of it's reach...get a bigger gun. I don't buy this ultra mag stuff...it takes 1 birdshot in the right spot in head or neck to kill a gobbler. All this extra high powered bazooka loads are overkill in what used to be normal spring gobbler hunting situations. It's just not my type of hunting to have the odds so heavily stacked in my favor..I like a challenge when I hunt. I alsodon't believe in wasting game..and have lost one gobbler in over 40yrs. of hunting...and he didn't go to waste.