I'm wondering why Hagrid isn't allowed to carry a wand or do magic.
We know that Hagrid carries a pink umbrella with the suspected remnants of his broken wand secreted inside (CoS chapter 7), and that Hagrid does do magic here and there even though he's not supposed to (SS chapters 4 and 5).
We learn in Chamber of Secrets that Hagrid was expelled from Hogwarts (CoS chapters 7 and 13), and that his wand was snapped in half. However, canon also states that witches and wizards are not required to attend Hogwarts; they can attend a different school, such as Beauxbatons or Durmstrang, or be homeschooled (DH chapter 11). To me, this suggests that a witch or wizard doesn't necessarily need to attend a school in order to become fully-qualified. While it doesn't expressly say so in canon, I note that the Weasley twins still use magic routinely after they quit Hogwarts (OOTP chapter 29). Why would expulsion from Hogwarts preclude Hagrid from learning and using magic?
Why can't Hagrid have a functional wand and be allowed to perform magic?