I used to argue this point with my father also and I agree that a perfectly tuned arrow with a perfect release and perfect form it doesn't matter. But, what I have come to realize is that I'm not perfect.
You can see in the video you linked that the clearance between the fletchings and riser or shelf is extremely small. It doesn't take much of a mistake in you release or form for this clearance to go away.
I personally have a quirk in my release or my form that causes a knock high fairly often even with a properly tuned arrow. This summer I tuned arrows to a new bow and it was shooting bare shafts perfect. I fletched a couple arrows with a 12:00 cockfeather and they shot broadheads perfectly. I fletched four more arrows and forgot to rotate the cock feather to 12:00. I went out and shot these four arrows with broadheads and got a severe knock high and all grouped 6 in low at 20 yds. I rotated the cock feather to 12:00 and immediately got perfect at arrow flight again.
I'm not saying you're wrong that it shouldn't matter, I just think sometimes cock feather orientation can help 'fix' or hide individual flaws in a persons form.