[Harry Potter] Getting expelled from Hogwarts

As I understand being a witch or wizard is something genetic; you are born one, start developing magical abilities at a young age, and it is something that you are. When you are expelled from Hogwarts, they break your wand, and can't attend the school.

But what is stopping you from going to Ollivander's and getting a new wand? Or to a wand shop in a different country where Hogwarts isn't the primary institution?

It's not like they remove your magical ability, Hagrid can still cast spells.

And then there are also children that are home schooled. They can't be expelled, so no matter what happens, bar something illegal, they can probably still legally do magic.

Is there really anything bad about getting expelled, other than social exclusion (eg not being able to get a job at the ministry)? You can still perfrom magic, and continue your education through self teaching.