Thanks for the chapter!
Not sure if artist only told father and son executed or something, but should be "hanged before the public." not "beheaded before the public." Beheaded means their heads were separated from their torsos by a blade. Like a guillotine, ax, or in rare cases a sword cutting through the neck completely. Usually a sharp blade is used, but sometimes to make the condemned suffer or incompetence on maintenance a dull blade is used requiring multiple cuts until finally get all the way through. Unlike Harry Potter's Nearly Headless Nick's killer, they need to make sure that if say someone is being beheaded they're beheaded as half assed, incomplete executions would make them look bad or can't even get a simple task done even if rep at the lowest and incompetent as hell.
Good MC and his family were spared. Most governments with kings, princes, etc. around that level tend to discourage treason by executing the family of the traitor (varies between just wife, kids, mother, father, or if include aunts, uncles, and cousins once or twice removed or anyone with any blood connection.) So has MC revealed that he's not just talented but the progenitor of the family (or at least person credited with being first and the family line starting with him?)
@Wolvenworks Depends on how the noose is tied and whether the intent is to give a quick death by breaking the neck or a slower, painful death by asphyxiation from the rope cutting off their ability to breath. Wire isn't usually used as it's not good for breaking necks, and if it cuts partway or completely through the throat the blood would make a mess that would have to be cleaned up to avoid increasing risk of disease and pests like flies.