In Scandinavia in the 17t-18th century, executioners were part of a sort of parallel society along with nightsmen (who emptied latrines), branded criminals, and so forth. They lived what was called 'honourless' lives, their wives were in the same class, their children and so forth. Having an executioner of nobel descent, even a royal one as depicted in Game of Thrones, was highly unlikely. Executioners, along with other honourless, lived life essentially completely removed from the social life of the city, they worked in.

Their jobs included both executing punishments, both physical and social, in public, and in many cities, such as Malmø and Copenhagen, they were responsible for a myriad of other jobs deemed honourless, or without honour in some way, like removing the corpses of dead horses from the streets.

The obvious question is 'why executioners and their families were branded as honourless' - several theories on this subject exist in the literature, but most prevallent is that dishonour, as much as honour, had an almost physical manifesation to it. When someone was dishonourable, they had an aura of gossamer-like dishonour about them; people treated convicted criminals who had not had their punishment executed with physical disdain, afraid that dishonour would rub off, as if it were contagious. Executioners had to live in this aura of dishonour, had to touch criminals, hit them, punish them, and worst of all, touch the criminals' blood. And such, they were a living embodiment of dishonour.

A point of interest that might help enlighten how executioners lived in Scandinavia is that many honourless people had considerable networks throughout Scania, Sealand, etc. We know this from massive criminal organisations that took root in these parallel societies. Basically, the whole northern part of Sealand and southern part of Sweden had mafia-like gangs of nightsmen and the like, with arranged marriages, underground dept-wars, family politics, etc.

As a small addendum, this sort of 'living-by-society' or living parallel to society was not exclusive to the honourless trades at all, the Jewish populations of Göteborg, Stockholm, Copenhagen and some other Danish towns had their own legal system, synagogues, clothing stores, drinking places, wine production for religious purposes and so forth. -- though that is not to imply that the Jewish societies lived any sort of honourless lives in any way compared to the nightsmen, executioners, and others branded honourless.

PS. As an avid fan of the asoiaf books, I never considered that Illyn Payne was from lower nobility. That is wild - very thematically poignant for the books, very unlike what would be practiced in the period you mention.

Sources:

• Krogh, Tyge. 1994: ‘Bødlens og natmandens uterlighed’ Historisk Tidsskrift

• Krogh, Tyge. 2000: Det store natmandskomplot

• Henningsen, Peter & Langen, Ulrik. 2010: Hundemordet i Vimmelskaftet

• Mührmann-Lund, Jørgen. 2016: ‘Good order and police - Policing in the towns and the countryside during Danish absolutism (1660–1800)’ Scandinavian Journal of History

• Rau, Susanne. 2007: ‘Public order in public space: tavern conflict in early modern Lyon’ Urban History vol 34, 1

Edit Added a section on why executioners lived dishonourable lives.