I think it's possible you completely misunderstood my posts?
Manually unwrapping is the problem I had, not wrapping. I think it is also a bit demanding, perhaps. Because FWIW, my unwrap tool is from an OK Industries WSU-30, at least as much as I can remember. It was off a double ended tool with a hexagonal bar (of I wanna say it was blue anodized aluminum?)* for the handle, and there was a wire stripper blade you poke the wire into. I clearly remember it was OK Industries, but I can't be certain of the model number. (Because I dismantled it for the bits, not liking the wire stripper). Anyhow, it had the tendency to grab two loops of wire and then just seize the tool as you twisted; you have to watch and carefully attain the end of the wire, only, or you basically cross-threaded things.
This is using standard pin header, which I measured out to 25 thous, which I understand to be the standard post for this wire. oops! see edit at bottom Kynar wire from Wes Bel (I think, or something like that?), which drop ships many components including this wire for Mouser.
*And that was before I went back and noticed the full details of your post. Went back to quote this:
The internal dimensioning of the tool is critical to get good wraps.
I can't be sure which one of my wrapping tools is from the Oki tool, anymore. IIRC, it worked pretty ok. I have wrapping bits from at least 5 different sources/tools, and I have compared their performance and viewed the wraps and bits under magnification. To me, the best is from the vintage Radio Shack wrap tools which used to be marked that they were made in Japan... near perfect tension of the wrap and no problems with double-wrapping.** The only thing that bothered me was that the first wrap wasn't tight against the board or header spacer. So I sanded a couple thous off the end of mine. This wrap tool also happened to have the most intricate parts with the most complex (I imagine) machining and the best finishing under the microscope. All others were a bit simplified in comparison.
I seem to recall some instructions on the original cardboard backing, perhaps, which said that it was also an unwrap tool. Instructions to turn CCW a half turn or whatnot to loosen the wrap. I found the wrap will then slide off, easily, in one piece. Maybe it doesn't work with all wrap tools.
**One of my wrap tools had this problem, because the internal bore was too big. It worked if you learned the trick, but if you didn't, you could get the wrap over itself. This was from an OKI tool that was a like a large blue plastic pen with a wrap bit for the tip, a spool of wire on the end, and a slide lock on the side. It is completely possible they were intended for a different kind of wire. I bought them used, and they indeed came loaded with a 30AWG wire with a non Kynar and significantly larger diameter outer insulation.
edit: Just dug out the unwrapping tool and tried it. And it seems like it works every time on standard pin header. I must have given up on it years back when I had wrapped a bunch of connection directly to thru hole IC leads.
Well, I put it back on my tool rack, lol.