I kilt my first birds many yrs ago with my Grandpa's old Winchester model 12 20 gauge, 2 3/4" high brass 4s. Then went with the 12 gauge with 3" magnum loads number 5s. Now I shoot the Mossberg 835 with the 3 1/2" 5s.
The 3 1/2" will punish you on the range for sure, but they do pattern quite a bit better out of my gun, compared to the 3".
As far as explaining the extra range or firepower. Well, if you hunt turkeys long enuff you'll come to understand that these wary birds are extremely tuff. If you don't break em down they can and will get up and run away. When I pull the trigger I want to be confident the bird goin home with me. I do not shoot at them at 50 yds tho, but that's just me.