Does anybody else think it’s wild that they break your wand when you’re expelled from Hogwarts?

There are so many reasons this doesn’t make sense to me…

For one, wands are a nearly universal wizard thing, but Hogwarts isn’t. There are other schools, and even children under Hogwarts jurisdiction don’t have to go. (How some students mention that their parents considered not letting them return for a particularly dangerous year.)

Also, not having a wand is pretty much removal from all wizarding society, unless someone super powerful sticks their neck out for you (Dumbledore with Hagrid). It would be like if, in real life, someone was expelled from high school, so the government took their car, destroyed it, took their drivers license and banned them from ever getting another one, and kicked them completely out of the country.

It seems like a really extreme reaction to being expelled from school as a child, especially if the means of determining expulsion are so often flawed. Hagrid was wrongfully convicted of killing a student; the situation could easily have been seen as, “Oh, this kid found a big spider.” But no, instead of getting rid of the creature, they get rid of the kid.

If the government can’t trust witches and wizards with a wand if they haven’t completed all seven years of wizarding education at this particular school, then they shouldn’t even allow the students to have wands when they go home for breaks and things. Following their logic, this would make controlling underage magic a lot easier, especially since the means of detecting it are so fundamentally flawed.

(As a footnote, do they destroy wizard’s wands when they get sentenced to Azkaban? Given their attitude towards punishment, I feel like they ought to, but I always figured the wands that Sirius and all the Death Eaters that later escaped had their own wands back. Do they hide them? Did the prisoners find them somewhere?)