UPDATE: Well, things are always interesting with technology…
I bought a $7 cheapie earphone/mic combo, tried it, and it worked. So, I got a new “proper” $30 Sony earphone/mic set, which also works.
So, it appears as though the old headset has an intermittent fault that decided never to show up while I was testing the stock Pie install before flashing to eQ and then eR, and only to show itself – each time, mind you! – once the new eROM was running. 
Sneaky technology! 
Funnily enough I also use the old Sony headset for video calls on my laptop, and there has not been a problem with sound or mic. So, maybe this becomes an X file (play theme music).
Meanwhile, the inability to make calls has been solved (admittedly, by buying a new headset), so in that sense this is now a “solved” thread.