Which mixer does a better job of mixing pizza dough, spiral or fork?
How important is incorporating air into the dough?

I’ve never used a fork mixer, always the spiral one. I think it’s pretty much an industry standard. How dough for pizza and any bread that you want to rise is like this. Mixing and kneading the dough works and stretches the strands of Glueten, the protein found in wheat. The more you stretch the glueten strands the better it will rise. What happens is a chemical reaction of sorts. The yeast feeds on the sugars and various other things in the dough, when that happens it realeses air and that air is trapped by the glueten strands which allows the dough to rise. Incorporation of air is more important with cake and muffin batters because you don’t develop the glueten that much and because instead of yeast you use baking powder.