Still, there are issues. Like, the prince's condition is not alarming but he needs to be nursed? What is not alarming about that?
Let's just say, you were experiencing muscle pain, and your sister helps you alleviate the pain by massaging it, that kind of feeling.
Why it's the girl who nurses the prince? There should be countless of servants to do that. Also, the King and Queen should have known of the situation through their spy network. It is their child and heir of the throne. Not to mention the gossips and the academy workers who would report to the King and Queen. You would really think what happens to the prince wouldn't get out?
Let's just say again, the academy is a boarding school. And they forbid the prince bringing lots of servant. He can't bring chef, stylisher, nekomimi maid, etc.
And the girl really cares for her brother, why not? Who dares to hinder a princess to care for her brother?
There is also the academy. Why not a tutor? After all, he is a prince. He could also have started magic training earlier since they have a lot of resources for that. Also, there would be various danger and enemies if he go into the academy, home schooling is much better.
A proverb that maybe can be applied in our modern concept.
The main function of the school is not for learning, but for socializing. Learning goes second.
They meet their candidate vassal here, also potential love interests (or political marriage tool) here.
And the king raised a heir worthy for his throne, not a NEET.
As for danger and stuff, oh well, let's just say the army / police / justice department is very skilled.
If some people dared to send assassin or something and they caught on it, every responsible party will charged with treason and their family got punished by 9 generations extermination.
Again, your excuse of them not telling the prince because he is a child. First, this is a really modern concept. Medieval times have children learn a lot of 'adult' things at the age we consider young. They have different concept of adulthood (like they train 7 year old on swordsmanship or get them to work on the farm). And even if we take the modern view: it is counterproductive in what way? I couldn't really think of anything counterproductive if the prince knows. In fact, in your story it is actually counter productive not to tell them.
Hmm, let's view this problem from a different way of thinking.
The information of figment princess is delicate.
The king for some reason can't disclose this to the public.
Can't he disclose this to his child who is still underage?
Of course no. There is no merit to do so. So what if his child knows or not, it won't change the fact.
On the contrary, it only increases the risk of information leak. What if he blabber this to the public later?
And what do you mean no defect is detected? It is a tumor . . . in the brain. Also, the King and Queen know of the abnormality but doesn't keep a monitor on it. Your child have tumor in the brain but then you just forget about it?
Add the word "other" between "no" and "defect" then.
All five sensory normal, the tumor is small and frozen solid, no growth and stand still just like accessory.
And it is in the brain area, even with magic, removing it would take enormous labor and high chance of failure.
The only one who can remove it effortlessly without any side-effect would be the girl herself in the future. (why? because the plot says so)
Let's just say, they did monitor, but after few years of no abnormality, they become lax with their monitoring.