Bob,
You mention that the children were seen by doctors constantly. I recall reading somewhere that they were seen at least once a week, and if I'm not mistaken possibly even more than that - as much as every few days! I wonder why there was such a frequency in doctor's visits? I can certainly understand regular visits to check on Alexei, as he was a child with a serious illness, but the girls were usually quite hardy and healthy, and yet despite the precautionary measures taken, they did get sick with such serious things as whooping cough, typhoid fever, measles, etc.
The children also led a relatively sheltered existence as far as socializing with other children their age, although I know that they were not completely isolated and did visit relatives as well as necessarily having daily interaction with the palace staff, all of which exposed them to illnesses.
Still, some may view these frequent, if not excessive, doctor's visits as the product of a well-meaning but very overprotective mother. They may also say that not even the most overprotective mother today who could afford regular medical care for her children would subject them to weekly doctor's visits if they are normal and healthy. Then again, child mortality was high during that time and diseases were rampant, so I can understand Alexandra's concern in keeping her children healthy. And I also think that these weren't lengthy exams but probably quick checkups. I imagine the children may have been annoyed having their time interrupted for these checkups, as most children hate doctor's visits, but then again they probably appreciated their mother's concern for their wellbeing.