I've noticed what I think is an issue with this post (I'm no mathematician though so this might be incorrect on my part). You say that Z+Z, or 1-1+1-1+1... + 1-1+1-1+1... = Z, or the original, infinite sum. When doing the math though, you end up getting Z+Z= 2-2+2-2+2..., which, according to the averaging-method used in the video to find that Z=1/2, would equal 1 since you would average 2 and 0. So Z+Z does in fact equal 2Z, or 1, and not Z, and this counters the rest of your argument about the "nonsense."
It's just my personal analysis of the explanation you gave, and as I said, I don't have too much experience in this field, so what I wrote might be completely false and I apologize if it is.