The Western European duelling tradition was developed in the medieval age by the warrior caste, the nobility. Women did not fight. The clergy did not fight (or was not supposed to at least). Commoners did not fight, they were in many places forbidden to wield weapons.
By the 19th century the real duelling tradition was dead, but it was aped by the growing middle classes, more as an affectation.
You're conflating the medieval code of chivalry, which governed single combat, with the later duelling tradition. You're correct in that duelling arose out of the earlier chivalric code, but there's an important distinction.
The code of chivalry did belong to a warrior class (knights), and it dictated the circumstances under which a knight could yield to an opponent without being considered a coward. It didn't really have much to say about affairs of honour.
The duelling tradition arose in the Renaissance as a means of settling disputes, supposedly
without bloodshed. Although gentlemen went about armed, duels were not supposed to arise on the spot. Rather, the code demanded that the party giving offence have the opportunity to apologize.
The distinction is that the medieval knight (as reflected in e.g. Le Morte D'Arthure) is expected to throw down at any time; the duelling code is supposed to do away with that. Gentlemen no longer carry weapons to murder each other; they carry them to mark their status, and to deter assaults by criminals (that is, for the same reasons gentlemen until quite recently carried walking sticks).
The irony of the duelling tradition is that although it was supposed to prevent bloodshed, honour codes dictated that one could not back down from a challenge, so it merely
formalized bloodshed and perpetuated the custom of killing people over things we would laugh off today.
For general consumption:
Here is a link to the southern Code of Honour, which was supposed to govern duelling in the South. It also includes the Irish Code Duello as an appendix. Note how ridiculously complex the duel has become. By the time this was published, duelling in the US was dying out.
https://archive.org/details/codeofhonororrul00wils
This link also provides the rules of the Irish Code Duello:
https://www.geriwalton.com/irish-dueling-code-or-irish-code-duello/
And this one, the French tradition:
https://www.geriwalton.com/french-dueling-codes-for-swords-pistols-and-sabers/
The most important thing to read on duelling, however, is probably this:
http://www3.amherst.edu/~cgkingston/duels.pdf
I've referred to this paper before, in this thread and in other discussions. It not only provides a rational basis for the duelling tradition, but it also explains how cultures of honour work, and why a gentleman who says something cannot lightly back down from his words by offering an apology. And it should make clear that duelling was never an affectation.
(And also why I object to the idea of bloodless duels.)
The idea of a "professional duellist" fits the model rather nicely -- a man who increases his credit by continually demonstrating his "honour."
Lots of good fuel for game sessions.
