Thank you.
However, while Googling to try and see how others had dealt with a similar situation, it was brought to my attention that there is a calibration option on the system settings menu. I didn't know this, probably because I'd never needed it. As you can probably guess, it has fixed my issue. Arguably, it would have been good if when I'd contacted Nintendo, they'd pointed me towards the calibration option, but never mind. Meanwhile, they sent me a very nice and apologetic email saying they'd arrange for a collection of the controller, for a refund or replacement. So now I feel a bit guilty for slagging them off.
This was slightly overshadowed by my new AV receiver arriving, and immediately exhibiting the same issue as my old one, i.e. the HDMI audio return channel not working. This led me to suspect it was the tv rather than the amplifier that was at fault. This suspicion was confirmed when I did what I should have done on Wednesday, i.e. connected the receiver to a different tv and tried that. So I sheepishly put the new receiver back in its box, and started plugging the old one back in, with a view to giving the tv a chance to behave itself before I contemplated how the fuck I'd get a warranty repair done on a 65" tv in the current situation, and indeed if I really wanted to part with it for weeks on end (as happened when Zelda hearts got burned into the screen). I got as far as plugging in the power and the tv HDMI cable, and tried turning the tv on. Imagine my surprise when the amplifier turned on too, audio return channel and all. Baffled (wondering if it could be to do with things being switched off/disconnected for a bit) but pleased, I set about reconnecting all the speakers and HDMI cables. That took a while, but no matter, I switched the tv on, full of hope. Hope was dashed, as the amp (just realised I've been using "amplifier" and "receiver" interchangeably; deal with it) remained silent. This could only mean one of two things: either an intermittent problem (which would be a nightmare) or something plugged in via HDMI was playing silly buggers with the HDMI CEC signalling. Praying for the latter, I began unplugging things. I have a lot of HDMI stuff - all eight inputs on the amp (and all four on the tv) are in use. Inevitably, when I unplugged the last thing (plugged into the front socket on the amp), the ARC and CEC started working. That last thing being... a Facebook Portal TV. The very thing that I bigged up on these very pages, just the other week. Well NOT ANYBLOODYMORE, MARK SO-CALLED ZUCKERBERG! But of course I was being hasty, it must be the HDMI cable? No, same with a different cable. How about plugging into the tv, rather than the amp? No, same issue. Just to confirm, I unplugged the power on the Portal, and after a few seconds, the HDMI audio/controls started working again. But then I plugged it back in, and it continued working. Basically, I spent £700 on a new amplifier, and came scarily close to waving goodbye to my tv for weeks, because the bastard twatting Portal had crashed! I have a good mind to take the fucking thing and stick it up Zuckerberg's arse, sideways.
So now, I have to apologise to both Nintendo (and refuse their kind offer of a return) and Richer Sounds (and ask them for one). My amplifier, controller and watch have all turned out to be fine.
Morale of the story: if in doubt, it's probably a software issue.