The purpose of "one" in "....one [insert name here]"

As mentioned above, the purpose is discursive, in that it indicates this person has not been mention before in the larger discourse–is being identified for the first time. After that point, the person will be referred to by just the name. I believe it’s common in legal discourse.

It’s similar to the very common practice of using the indefinite article “a” before referring the name of someone unknown to the speaker to a third person.***Employee: Did I receive any calls when I was out?
Receptionist: Yes. A Mr. Robert Smith called. ***The receptionist uses the indefinite article “A” to indicate that he or she is not familiar with Robert Smith, though the employee may or may not be familiar with Robert Smith.

I don’t know about that. I find many (relatively recent) examples which are not from law enforcement which don’t appear to be ironic:

Southern Living Mag - 1995
Vol. 30, Iss. 11; pg. 100The revival of the Olympic Games, Athens, Greece. America’s " patched-together team " mistakenly arrived just a day before the Games began, forgetting the 12-day difference between the Greek and American calendars. That same year, a British tourist, one John Boland, casually signed up for the tennis tournaments – and won.
Blog, 2010
"Making Light: A different kind of ‘political science’"The convention (unwritten) was that the premier would replace the deceased senator with one from the same political party; however, since this is not specified in the constitution at the time, there was nothing to stop the then Premier of Queensland (one Joh Bjelke-Petersen) from filling the vacancy with a member of a different political party (ie not Labor)
NPR Radio, 2002
Profile: "Street Kids of Mumbai"And pretty soon when they’ve got a steady income of their own, they settle down to a hut of their own. One Sadak Chaap on P D’Melo Road’s just got married to the daughter of a pavement family and gone to live with them.
Glee (Ep: “Britney 2.0”), 2012Brittany, I’m afraid it’s time for a little tough love. I hold in my hand the most recent algebra test of one Becky Jackson. Our plucky little Ewok waddled her way to a respectable C-plus.
Today Show, (5/28/2015)
It’s now home to the new Broadway hit An Act of God, a comedy where the almighty takes over the body of one Jim Parsons and addresses life’s great mysteries.
Colorado Springs Gazette, 4/03/2017
“The wonder of might of Colorado’s Great Sand Dunes”The dunes watched as cattlemen and sheep men waged wars in the surrounding fields, watched as prospectors came to dig for gold (the dunes were suspected to conceal fortunes). The greedy fever compelled a valley woman, one Elizabeth Spencer, to rally up a conservation effort, according to historical accounts.

(Maybe the one from Glee is ironic. I haven’t seen the whole episode.)
There are many more, but I don’t have the time to list them.

These days I see it mostly used ironically. Anyone who’s likely to see Birds of Prey knows full well who Harley Quinn is.

Would you have any examples of it being used ironically (apart from this one example)? In the examples I find, I see that:

  • It’s used more in fictional writing than in other contexts.
  • In fiction it often is part of dialog of characters involved with law enforcement or crime, non-ironically.
  • When it’s used in non-fiction, such as in newscasts, etc., it’s not used ironically or for humorous effect–it’s used “normally.”

Of the many fictional fictional examples I find, I think you can say that–being fiction–it’s used for a more “dramatic” effect.

But of the non-fictional examples, here are some more which I think one could say are clearly not ironic:

From the academic journal Humanist, Jul-Aug 2009, Vol. 69 Issue 4, p23-27, 5p:
“Counterfeiting Tradition”
" He describes the evolution of one tradition, which in original form read: " The Prophet gave the order to kill all dogs except hunting and sheep dogs " Later it was changed by one Abu Hurayra, who added the words: " but with the exception of farm dogs as well " Al-Bukhari wryly commented, " Abu Hurayra owns cornfields, " that is he had a vested interest in amending the order so that his farm dogs would be spared.

From Fox network, Brit Hume, April 14, 2005:
"That is Dianne Feinstein today, explaining her vote in favor of one Thomas Griffith, a man we’ve discussed earlier on this program, to be a member of the Circuit Court of Appeals for the District of Columbia, an important and perhaps the most important of the appellate courts.

From a blog, “Charlies’s Diary,” Oct. 27, 2020:
“. . . who’s to say the Okhrana is not to fund some local offshot Mahdists, giving one Nikolai Kibalchich ample oppurtunities to sublimate his revolutionary urgings and see some of his designs in action with plausible deniability, with Quing China acting as a proxy;”

From Omni Magazine, Winter 1995, p. 58
“The Secret Invasion: Does It Add Up?”
"Meet one Katharina Wilson, an attractive, intelligent, apparently well-adjusted, 34-year-old woman. Born in a small college town in the Deep South, Wilson now lives in Portland, Oregon, with her second husband, Erik. "

From Huffington Post, 2012
“Paul Stoller: Joseph Kony and the Other Africa”
“The continent of Africa is once again in the news. This time the buzz is about a viral video, " Kony 2012, " which is about the unspeakable atrocities committed by one Joseph Kony, leader of the Ugandan Lord’s Resistance Army, which has consisted of a ragtag group of dazed and confused children who have been kidnapped.”

Sure. Here are a few other ironic uses of “one” preceding very well-known names:

“…in the personage of one Donald Trump and his norm-shattering rise.”

“…viewers were likely stumped to see one Tom Hanks in the opening credits.”

“Enter again one Hillary Clinton to grace us with her thoughts…”

“…not limited to the excesses of one Vladimir Putin, who took a few years off…”

“…prior to making the decision, he called the actor’s previous director, one George Clooney…”
(Why George Clooney Doesn't Really Act Anymore | Cinemablend)

“President Trump will be meeting with one Kim Jong Un during his trip to Hanoi…”
https://www.nydailynews.com/tags/kim-jong-un/

Right, well obviously if it’s used with a famous person, it has to be ironic–but that’s ironic about the person. What I’m trying to find are cases in which the construction itself is used ironically. So, used not with a famous person, but still self-consciously. In other words, is the construction itself essentially archaic, as the OP suggests.

My OED lists the use of ‘one’ in this sense [1] simply to mean “a person; some one” as archaic or obsolete, as in “Then one brought hym a cup with wine.” / “A mad dog had suddenly tore in pieces a garment about ones body.” / “This month one went by land to Connecticut, and returned safe.” / “One in the Hundreds of Essex made a great improvement.”, but makes no such claim of obsolescence for the use we are discussing (“defined by a noun in apposition”) or when “defined by a clause or phrase”, as in “Mr. Carlyle is as one who does not hear the question.”

[1] A person or being whose identity is left undefined; some one, a certain one, an individual, a person (L. quidam).