You are right on a certain point on the fact that boys are generally not enough mature to care about their health until the last grade of high school as long as they are living at the home of their parents and are treated by them as children.
So I agree that there is nothing wrong if a mother is coming with her son when he went to see a doctor.
I remember that in my case, she was ofen coming with me, even after 14 age if I was sick and if she was anxious but it was never for a complete routine check up since I had one each year and rather thorough at school, so I went to see a doctor only in case of a special sick, And the doctor who was a male did not perform a complete exam and I had generally not to undress completely, just a few times, not often, in briefs.
And my mother was sitting face to the desk of the doctor and I believe that she did not turn her head to look at me when the doctor was performing his exam at the back of his office. She was very dicreet and never talked to me about my undressing, she was only interested by the result of the exam and the advices of the doctors.
I remember only one time, certainly aged of 14 or 15, when the doctor wanted to look at my genitals, so he told me to go with him at the back of his office and told me to undress into briefs, after a visual exam of all the body, he told me to pull down my briefs to look at my genitals, he lifted my foreskin and he also palpated my balls, then he told me to come on the side to be weighted on a scale ( in brief ) and then he told me to turn around to inspect my spine, he slided his fingers all along my spine from the neck to the cleft of the butt, he lifted the waist of my brief from back just for a short look but did not pull it down below the butt.
I was not facing my mother, so I cannot be sure that she had not turned her head to see the inspection of the doctor but even in that case she had not seen a lot of my body, anyway she had not talked to me about the exam and my undressing.
Later when I was a university student, she came with me just one time for an issue on my balls, after some questions the male doctor instructed me to strip into briefs and to lie on exam table and he pulled down my brief to look at my genitals and palpate my balls, my mother was sitting not far from the examination table but I cannot know if she had seen my balls being palpated , anyway I was myself anxious, so I did not care if she saw me, she was my mother..and being anxious about my health, I was not embarassed .