So I'm really delighted you (or your patreons) chose this story, even though, ultimately, I agree with you both that it doesn't really deserve, on its merits, a place in The Best of Gene Wolfe. Because I really, really enjoyed it — the story, and rereading it, and the discussion, too. I don't know why; maybe it is precisely the lack of an SF element to the tale, save for the meaning in the world suggested by the record having the meaning that the narrator imagines in it — even if not in quite the sense that he imagines it. I think it's well written, well organized — a delightful story.
Among many other things, I loved your take on why Wolfe included it in the Best-of: which I think is right. His afterwards is really quite something. (I haven't read most of Best-Of, since I have all the individual collections, but then I miss the afterwards unless I catch them on WolfeWiki or quoted by Aramini or something...)
(By the by, you mentioned a fan-edited Best of Gene Wolfe.) In that volume — in the afterwards, perhaps? I don't have it handy — Wolfe suggests a second volume could be put together of fan-edited stories — the suggestion being that it is in addition to the ones in BoGW, so they would be, so to speak, out of bounds. In his introduction to the UK edition (reprinted in the US in a past issue of The New York Review of Science Fiction), Kim Stanley Robinson notes this comment, and lists a bunch of his own suggestions — including, of the ones you've so far covered, "The Changeling", "Eyebem" and "The HORARS of War". He also suggests '"A Story', by John V. Marsch". Anyway, if you (or anyone) does a fan-edited volume, I would propose it be that one: a companion, non-duplicating one.)
As far as the mystery of the story goes, I agree with your reading — which is also Marc Aramini's — that the narrator dies at the end. I agree with all of you that it is, basically, obvious.
But not so obvious that people can't get it wrong; and hence the one thing you both misinterpreted I believe — not anything of Wolfe's, but Asimov's comment.
Asimov's comment is from the book 100 Great Fantasy Short-Short Stories (1984 — so not right after the story was published, as one of you seemed to imply), co-edited by Asimov, Terry Carr & Martin H. Greenberg. First, each of the stories got a single, one-line comment from Asimov — a short intro, I think, to match the short-short stories. So that was not a sign of disrespect, I don't think.
But it was a misreading. Asimov was, of course, a noted rationalist; he was also not (to judge by his self-characterizations in his own work) much of a careful literary reader. This story may well, who knows, have been selected by Greenberg or Carr, or selected by Asimov from a set they chose, or something (he says somewhere his contributions to anthologies tend specifically to be the introductions). At any rate, I would suggest that Asimov quite clearly read it quickly, not very astutely, and thought that the last line was meant to be taken straight — that it was a send-up of superstition. Asimov, basically, blew the story.
So why did Wolfe highlight it? Who knows. He could be poking gentle fun at Asimov; he could be slyly trying to encourage his readers to think more deeply. He could have just found it funny. Or all of the above.
Why did F&SF publish it? Why did Asimov, Carr & Greenberg reprint it in a book of fantasy stories? Well, read properly (i.e. not as Asimov read it, but as you both and Aramini did) it does have a fantasy element, albeit a slight one, insofar as it presents the world as meant, things as signs having a meaning that science does not strictly imbue them with. And yes, Wolfe is a name — although rather less so in 1972 than, say, 1984, or certainly now. But it may have been something else. There's a genre of SF/fantasy story where something mysterious is teased, as either being supernatural or not; the X-Files story. But unless you print some red herrings, then readers always know what the answer will be. (This is a problem in X-Files; in the original conception, Scully was supposed to be right half the time, not how it turned out, obviously.) This is a perfect red herring: since at first it seems to be one, but carefully read, it's ambiguous, or perhaps better say liminal.
Finally, clones: it's absurd, as I rather suspect you both know, and I shan't bother to argue against it. Obviously genetics and family traits and all that are a big part of the story, but one needn't go all the way to clones for that: simple family resemblances will do. I liked your reading of the "stamped" line as having to do with the record — but, again, you can get that from ordinary genes. It reminded me of the early sonnets of Shakespeare, where he emphasizes that you ought to have kids so your looks will live on. (Sonnet 3: "Thou art thy mother's glass, and she in thee/Calls back the lovely April of her prime:/ So thou through windows of thine age shall see/Despite of wrinkles this thy golden time./But if thou live, remember'd not to be,/Die single, and thine image dies with thee.") — Though now, of course, we need a story about how Shakespeare was really talking about Clones.
Again, it was really great to read a (comparatively, for Wolfe) story, and to hear you talk about it. I particularly loved the idea that the feeling of power line was the answer to the riddle of the 'I thought I loved him' line: just right. And the Proustian memory bits were great preparation for Fifth Head, which I am greatly looking forward to — as I am your recap episode as well.
(PS: I suppose it's too late, what with y'all living in the future, but do we have to call Fifth Head a novella collection? I mean, sure, I guess it's that too in some sense. But really, it's a novel — Wolfe's first great novel. Let's let both it, and the novel form, have the dignity of the ascription.)