Problem is fixed - I repeatedly and vigorously cleaned the 510 inside and out with a toothbrush, and I also sprayed alcohol onto the underside of an atomizer and screwed it on and off about 15 times in quick succession. Even that didn't fix it immediately, but after a few more atomizer screws/unscrews while doing further testing, it started working consistently. It's now picking up the right resistance with Set Resistance and vaping in TC mode without ever showing SHORT. I'm going to clean it a few more times just to be absolutely sure, and won't put it back into my 'vaping rotation' until tomorrow
But to address your points:
A short would not measure 0.00Ω, it would measure the resistance of the material causing the short. Even if you wired the positive on the board directly to the board ground, it would not be 0Ω - it would be the resistance of the length of wire between those two connections. Which true in that case would probably be so low as to display as 0.000Ω on the mod, but would not actually be (a short section of copper might be around 0.0001Ω)
In the case of this short, it was being caused by dirt or other material - not a solid connection. So an ohm reading of 0.3Ω - 0.7Ω as I variously saw is quite believable; it's comparable to a very poor/loose connection on a coil wire. There is a connection there, but the ohms vary and are much higher than a solid connection.
Perhaps you forgot that when when I used my micro ohm reader to measure the resistance between the positive and negative of the 510, I saw a resistance - and its value was in the region of a value I had recently seen with Set Resistance. That proved beyond doubt there was a short on the 510: a 510 without an atomizer should not give a resistance reading; it's an open circuit until the atomizer closes it. Now that it's working again, I am correctly seeing no resistance across the 510 when no atomizer is attached.
As for "the only thing that functions differently in TC mode is the board. the 510 is static." That's exactly right - and that's the point. The 510 is static but the board operates differently in the different mode. So the cause of the problem (dirt/crusted juice/something in the 510 causing a short) was always there - or rather, it was always intermittently there - but it was not causing a visible problem in Power mode, because the board operates differently in that mode than TC mode. Crucially, it does not have a locked Set Resistance value and thus does not report an error if the ohms suddenly differ from that locked value.
I believe the key to the issue was that, latterly at least (after I started posting and by the time I tested Power/Kanthal) it was only intermittently shorting. By that point I had already done a lot of cleaning, and after I had cleaned a lot there must have been only a small amount of dirt remaining such that it was sometimes shorting, sometimes not. It was shorting when I measured the resistance across the 510 with my ohm reader, and it was on the occasions when I got a Set Resistance reading with no atomizer attached. Then while testing, when I did Set Resistance I sometimes got the resistance of the short, sometimes I got the real resistance of the atomizer. Most importantly, during TC vaping the short occur sporadically, and whenever it occurred the board aborted the vape with SHORT. In Power mode with Kanthal, that was no problem - it carried on firing based on the initial resistance, not caring if there was a short for 0.01 seconds. In TC mode, it caused the problem because of the board's short detection: as soon as it picked up a very different resistance than it had locked, it aborted the vape with SHORT.
So yes, same 510, same underlying problem, but different behaviour by the board making the problem apparent in TC mode and not in Kanthal. Maybe if I had properly vaped for some minutes in Power with Kanthal, I would have noticed the vape dropping off intermittently as it suddenly got a much lower resistance reading and cut the volts. Though if it was only happening for a fraction of a second at a time by that point then it might have been very hard to spot.
Anyway it's all OK now. Thanks for your help.