There was, in the end, precisely no mystery to the faulty modem. All it required was a firm push in the right direction (down) to make it go. So, while I’m happy that it now works, I also feel like something of a doofus. I love that my phone has now been taught the word doofus.
I used to enjoy looking at the user dictionary in any word processor I’d been using for a while – I revelled in the swears and the made-up words.