The guy cutting off his head did not know how to do it properly and made an embarrassing botch of it. Had they shown the whole thing, we would have seen the amateur cutting and cutting and sawing and getting nowhere. He may have even ended up cutting himself pretty badly in the process.
I have seen other examples of exactly this problem. Apparently it is pretty difficult to cut off a head and it takes a lot of effort and quite a long time.
It doesn't produce the sort of drama they are expecting, and makes them look foolish. I have no doubt that they originally intended to show the whole thing, but they did not reckon with how hard it was.
Anyways, that's my theory.
Maybe they should have been watching Dexter.