Actually, when aristocrats were beheaded in Tudor times, many of them paid their executioners to be sure the blade was sharp enough to do the deed with one clean cut. Prior to that, beheaders would just hack away at the person, which makes me quite ill.
I guess this is why the guillotine was considered so innovative. WHAP -- and there went the head!
As for the question about beheading Quentin publicly, I believe the rationale for that was that they were reverting to laws upheld in the 1600s. It's kind of flimsy, yes, but it kind or explains how they would even have a witchcraft trial in the first place, let alone a judgment of public beheading!