Several issues here.
The privacy aspect never occurred to me, but it actually makes some sense. Especially if I’m at, say, the VD Clinic, I probably wouldn’t appreciate some nurse announcing my full name to the waiting room.
The mutual respect aspect. Many doctors, frankly, take themselves very seriously and would never dream of allowing a patient to call them “Jim”. If I’m dealing with that sort of person, then I’ll also take the dignity of a “Mr.” when I’m being addressed, thank you. After all, I’m paying, and I’m being addressed by a well educated (and dedicated) tradesperson. Don’t even think about patronizing me.
(Now, my doctor and I are on a quite friendly basis, and while I actually call him Doctor, because I’m a traditionalist, I could call him by his first name if I wanted to. I wouldn’t want him to address me any other way, at this point. He knows all my secrets.)
The elderly-patient issue: Yes, these ignorant, cheeky, inept girls (of both sexes) in the front office should NOT be calling Mrs. Meriweather “Margaret” under any circumstances at all. Mrs. Meriweather winces inside when she hears this, which makes her sciatica much worse. See, no one ever called her by her first name in her life except dear departed Mr. Meriweather, who called her Meg. But certainly never in the earshot of a stranger. These clueless employees need to give her the dignity of her age; she’s earned it.
I’m happy to report that in the South, in nursing homes, they still do it right. I hear Miz Johnson this and Mr. Smith that from the young staff, unfailingly, which is a good thing when they’re talking about my grandmother, or they’d hear about it from me. It’s simple respect and professionalism.