The half has nothing to do with averages.
It has everything to do with integral calculus.
Start with momentum, MV. Integrate it once (with respect to V), you get 1/2 MV squared. If you integrated it again, you get 1/6 MV cubed, but this item means nothing to us, so we don't do it. (There are a few constants thrown in as well, but we won't worry about them either.) Purely mathematics, no averages.
