They are constrained by their low rank/not being recognized as adults, and have to compete to earn certifications to make progress towards becoming full-fledged warriors considered as equals by their teachers.
I wouldn't mind if you left that trope out! I think there's plenty of the "Oh waaaaah! we're so underestimated, misunderstood, and overlooked! But we'll prove them wrong and save the day and earn their respect!" kid stuff. [Adults are useless, Not now kiddo, Cassandra truth, Parental obliviousness, Police are useless, etc...]
Unless you are playing it for laughs and lampshade it, like Lemony Snicket (who did a fantastic job at that), it just gets tiresome to me.
I don't mind the students being hindered by their genuine incompetence and ignorance though!
Personally, if the adults aren't helping them with some challenge, it's probably part of their training. I'm a fan of seemingly (or actually) heartless wizard masters who toss their own pupils into different dimensions or labyrinthian dungeons, or teleport them to foreign lands to force them to survive. SERE and worse. I've heard offhand about survival courses like escaping from helicopters that are intentionally "crashed" into freezing water, or being dropped nude into a forest with only a hunting knife and expected to survive for three weeks before you are picked up (not sure if that's a real thing or not).
I'd expect the teachers to be harsh, dismissive, and abusive.
why this school exists
If a magic school existed in real life, maybe it'd fall into one these categories: (or maybe a mix of them!)
A) Military institution (either pre-military preparation, general soldier (bootcamp), or an officer school (e.g. Westpoint))
B) Correctional training
C) Shelter and rehabilitation (also think post-Nazi rehabilitation of concentration camp victims)
D) Private or government general education (Harvard, Yale, Hogwarts, etc...). Likely for ran for money or prestige or philanthropy, though maybe ran as a cover to hide a more important purpose,
E) Also see the different purposes for different types of concentration camps for more ideas.
Are these government owned but ran with minimal government interference, privately-ran and privately owned (private military), or government-ran and government-owned (and possibly overly bureaucratic)?
Then you have to ask, "Why was it originally founded?", "How long ago was it founded?", "How has it changed since the founding?"
As each general passes, the reasons for it a school to exist, and the purpose and motives of the people running it, change dramatically.
I want to have a more explicit process of specialization, where the teens have to decide what role they want to specialize in, then earn it, and there would be some kind of irreversible metamorphosis.
I like this idea. I read a poorly written (i.e. no editing process) snippet of a story recently that kinda went in this direction, and definitely wouldn't mind more of both those ideas (explicit specialization and character metamorphosis).
I'm aware that jedi in the Star Wars universe are kind of like ninja, but I never liked the philosophy or culture of the Star Wars universe much; I see jedi as being like paladins or clerics, and I strongly dislike religious warriors or mages.
Not even when they are genocidal eugenics?
I linked that for additional creative thinking; not to encourage or discourage the use of ninjas, clerics, or clerical ninjas.