FWIW
In case you were to take me up on my offer, I wanted to make sure I could come through - So I just cut some 12-24 threads out of 0.25" diameter O-1 drill rod, which I turned down to 0.2160".
I did it both ways - first by single-pointing on the lathe, then by using a "re-threading" hex die.
Both methods produced acceptable threads, though I prefer single pointing since you can control the fit.
But what surprised me was how well the die performed, considering this discussion. I wonder if your 12-20 die is dull?
Anyway, here are a couple of photos of this morning's activity.
Here is the hex die I used. It came out of a vintage Craftsman set I have had for years. (Sears sold them as thread cutting dies, but many claim they are acceptable only as thread chasers.)
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Here is the tailstock die holder. I used it under power, with no problems with engagement or with cutting.
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Here is the single point threading set-up
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And here are the threads. Not pretty, but they work.
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