There was no lawful authority where they were at. No town guard, no one with a legitimate sovereign claim on the land and certainly no representative of any supposed law was present. Where would they take him? If Neverwinter claimed the land then where are their law enforcers? If you claim it the you should administrate it. Phandelver would have said what do you want us to do about it? In the American frontier where their was not yet any territory government, frontier justice was not uncommon. The man was clearly guilty of attempted murder and conspiracy to commit murder. Some people are trying to apply modern law concepts to an unmodern situation. Yes they were judge, jury, and executioner because there was no judge, jury, or executioner available. The lawfuls in the party applied common law to the situation. Keep in mind that long term incarceration in a prison simply was not done in Medieval times, few places even had facilities for that. And certainly not for common criminals. For many crimes that we would send people to jail today they executed people for in medieval times. The alternative would have been a large number of lashes and then dragging him around until he could go on his own, healing on his own since the pain of healing was part of the punishment, but that would be for lesser crimes such as stealing.