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There's no evidence that it's painless and there is evidence that you stay awake and aware of what's happening to you for several seconds after you are beheaded.

We have no means by which to determine that death by beheading is not painful.

We do, however, have plenty of examples of nitrogen hypoxia, it's effects and the sensations associated with it as you die, by people who were either revived or recovered from situations caused by nitrogen hypoxia.

Personally, I would be okay with death by nitrogen hypoxia, if I either wanted to, needed to, or were forced to die before the end of my life naturally. I presently have no desire to die, nor any need to, nor have I been sentenced to death (or any sentence) for crimes (of which, I have not committed any).

So my opinion is just that, an opinion. I would vastly prefer to continue living at the moment; so I'll just stay out of trouble with the law by doing the same things I always have, and hopefully my health doesn't cause the situation to change.

Death by beheading doesn't sound very nice, but bluntly, it's hard to screw up with something like a guillotine. Since law enforcement (specifically those in charge of executions), seem to be inept, the guillotine may be a better option, since it would be much more difficult for them to do in a way that's so incorrect that it causes more suffering than what is normal for that process. IMO, that's the only significant merit to something like the guillotine. It's so basic they would have to try, in order to do it wrong.