I think it has changed over time, and can be regional or urban/rural. I gather where I lived was a throwback to an earlier generation. My last exam with my mom watching was in 1992, and I was in a suburb of a second tier city in the South. Also my pediatrician was pretty old, nearing retirement. So I may well have been getting the 1965 treatment.
The closest video I've found to my experience indeed is from 1960, so this likely is how my pediatrician was trained:
https://www.youtube.com/watch?v=oI4q-hGSRL4
The only differences in my case were that instead of wood paneling, the exam room had nursery rhyme themed wallpaper, and the height and weight in just my undies at 6:00 in the video were done at the scale down the hall and by the nurse, not the doctor.
Because of the editing, you need to pay special attention at 9:45 to see that this exam is just like mine were. First, you see the mother's sleeve at the left of the frame, so you know that the boy is looking at his mother while the doctor has pulled his undies down to his ankles. You also can tell that the mother has a very good view of her boy's penis and testicles during the exam. While the editing left out the part where the doctor pulled down the boy's undies, it kept the part at 10:00 where he pulled them back up. Fortunately, there was no tickling during my exams.
Finally at 13:15 it showed a page making clear that this type of exam should continue until age 18.
Obviously, this isn't how these exams are done anymore, but I'm glad I found this video as evidence I'm not remembering wrong.