No, I wouldn't show my kid that video, and I would be PISSED beyond belief if anyone (including my dp) showed that to my 13 or 9 year olds, no less my 6 year old!

We are not vegetarian, but the video was not appropriate IMO because it is inferring that a chicken running around with its head cut off is funny. There is no reverence; I think that is the offensive part.
I agree with limabean that the kid was asking for info about an expression in the English language, and I would explain it in the "hurried" way.
I think it is fine for kids to learn that dinner didn't originate in a plastic-wrapped package at Safeway, but with reverence not laughter.