Broke out the tools and my old Weller TCP, and set about replacing the resevoir capacitor on the black-case set. Leon's instructions worked perfectly, many thanks.
Result: no effect. Still faint signals and some hum.
Fitted replacement antenna while I had a think. Realised I could put together some D cells if I raided the Maglites, so tried that. Then switched it to battery! Same faint signals without hum. Short rectifier pulling voltage down? All I can think of for the moment. EDIT: mains circuit does appear to be completely isolated when in Battery mode... I'm tempted to disconnect the back of the good set and tag it on, to see if a known good supply fires it up, but I'm also a little leery of ending up with two dead sets...further edit...fault probably on main board somewhere - if PSU id fully isolated on batteries, then substituting the known good PSU won't make any difference either.
I'll get there!