A lot of what you describe is common to the era, except for all the touching of your genitalia, which does seem a bit odd.
But I had mostly the same treatment aside from that at different clinics. I was always told to strip to my underwear, and the door was closed. I could never figure out why, because once the door was open after that I rarely got much privacy. Once I was in my undies, the nurse would take me down the hall in nothing but my underwear to be weighed on the scale. You would pass other kids in the hall and there was no concern for anyone seeing me in nothing but skimpy white briefs. Once back in the room, if I was lucky the door would be completely closed, but more often it was only partially closed. Then it was hop up on the table for blood pressure and pulse. While she held my wrist I would be nervously sitting there staring at the jar of red tipped thermometers.
And then it was time. The nurses used baby talk with me as well, often calling me sweetie or honey when telling me to lie on my tummy and slide my undies down for then. They would often warn me it would be a little cold or tell me it wouldn’t hurt like I didn’t remember from the last time they took my temp. Then once it was in I got called a “good boy”. From that point they might talk to my mother would holding the thermometer in my bottom. They would ask her about my symptoms or other things like I wasn’t there. If I moved or said anything I would be told to be still or hushed.
Fortunately, though, the doctor rarely if ever checked my genitals or had me remove my underwear, except at 13 getting a rectal exam.
At the end of every visit the nurse would return with a needle on a metal tray, that always went in my butt. So the whole baby talk, on your tummy and undies down would be repeated until I got older and was allowed to lean over the table. I always got told to be brave or something along those lines, and got a toy or candy after, even if I screamed my head off.