That's not evidence for the existence of magic binding...and in fact is evidence that magic binding isn't a THING in Harry Potter. That's not even getting into how stupid the practice would be. Remember that what can be bound can also be unbound and all it would take is one Smart Dark lord willing to do so and they would have a ready made army of loyal minions.In the case of Newt it is explained that a person expelled after they completes their OWLs to be deemed good enough at controlling their magic that their wand isn't snapped, implying that the reason the wand is snapped before that is because they're considered a danger to themselves and others.
As for Hagrid he kept the pieces of his wand and was able to cast with it but he mentioned he was forbidden from doing so, although that was lifted after he was absolved after CoS, so I guess part of the reason he was allowed to be employed by Hogwarts was as a way to keep an eye on him.
Although with all this talk about Expulsion, i have to say as someone with a bit of an insider view into the Administration side of Education that Expelling a Student is not something done lightly. It is seen as an action of last resort and carries a bunch of legal issues alongside it(in this case Hazel may be breaking School Rules by not attending class but she isn't doing anything ILLEGAL persay). I suspect even in a backwards Wizarding World Expulsion is not something they want to be doing with regularity...otherwise your just creating a growing number of malcontents that only want to hurt the establishment.