"surferboard" :
it is a little surprising that your mother did not tell to your sister to get out of the exam room when you reached the age of puberty development
Normally it was to her to decide if your sister could stay or not during your examination.
Since she told you to get out when it was your sister turn to be examined, she could have told her to get out for your exam ..
The doctor had no reason to protect your privacy more than your mother. Since it was fine for your mother, it was fine for him, it was to the parents to decide..
Of course it was more embarrassing between 12 and 16 age than before, and personnally I think that it could be admitted when you were only aged of 12 and 13 but not between 14 and 16.. So if I were you, I would have asked at my mother and the doctor if she could leave the exam room before I undress when I was 14 age.
You had nothing to loose to try to have more privacy, even if your mother would still be present.
May be you could have told them that you wanted that more privacy towards your sister ( what was her age, was she older or younger than you?) and that you were embarrassed by her presence and you would have seen how they would have reacted.
Not sure that they would have agreed but who knows ,
Since it was your regular pediatrician for a long time ( you said that you went to an other doctor only when he retired and it was when you began your college years, so after 18 age.), you might not be too afraid by him...
Of course they could have answered to you that your attitude was stupid for a boy, that all boys were build the same and had nothing to hide, especially to siblings, and that your sister had seen you every year and that there was no reason to change the set up of the exam because you were becoming too more bashful , or that a sister had to learn about male anatomy which needed that she stayed in the exam room for your exam but that she was not interested by your privates except from a medical point of view. So it was not at all a big deal if she saw you naked.
And in that case, you had nothing more to say!
I think that your doctor was certainly an "old school" doctor who did not mind at all about the feelings of a boy when he was examined and though that a boysdid not need any privacy.
He was well known for his methods of examination since you told that some of your friends had stopped to go at his office because they wanted more respect of their privacy.
It happened often with old doctors or doctors who were retired military doctors who were acting a little like if they were still in the army examining draftees or new recruits, especially with boys and male tenagers, because they were used to perform examinations in the nude previously and to perform their examinations without thinking to the embarrassment of the guys, so they did the same with boys and male teenagers.
On the reverse, they were respecting the privacy of girls because they had not the same experience previously.
Anyway, your doctor was thinking that girls needed a certain level of privacy for their examinations since he agrees with your mother when she told to you to get out when it was the turn of your sister to be examined just after yours.
As regards the double appointment by your mother, it was very common in family to win time, it was the choice of your mother but I presume that the doctor agreed with it. May be it could have stopped between 12 or 14 and 16 age but it was more practical for both, your mother and the doctor!
You said that there was no partition or curtain and that you were examined with your briefs pulled down below your knees facing your mother and sister with arms on your sides, so in plain view of them when he was checking your genitals, first checking the small amount of hairs in that area with comments, then palpating your testicles, checking your scrotum, and lifting your penis, inspecting the shaft and retracting your foreskins several times, checking your urethra opening and eventually performing the check of hernia.
I presume that it was embarrassing to have his genitals exposed in that way to the view of your sister at 14, 15 or 16 age but it could have been worse : same kind of exam between 16 and 18 before you graduated from high school, still in presence of your mother and your sister!
Or much more worse, the instruction during your examination between 12 and 16 age to remove your brief and to stay stark naked for all the examination (or even just a part).
If it were for all the examination, you would have been weighted stark naked on the scale with buttocks in plain view of four persons : the male doctor, his female nurse (since if I had well understood she was also in the room ?), your mother and your sister.
And the instruction to walk in the nude in the room to check your balance, then to bend over and touch your feet to check your spine and why not some comments to your mother about your spine and your butts, if they were hairy or not after 13 age and why not his hand on your cheeks to spread them to have a visual view inside your crack and let your mother see by herself how much you had hairs in that area or not ( and sister of course if she sat near your mother), I think that part of the examination would have been also embarrassing than that of the genitals.
And for the genitals, it would have been easier also for him if you had been totally naked.
So it could have been worse..