Why there's GOT to be more than three magical schools in the HP world

So I spend kind of a lot of time on Tumblr, and it's impossible to do that without coming across A LOT of Harry Potter headcanon. A combo of my own particular love of wider worldbuilding, regional fantasy from all regions, and talking at length with my niece who just finished reading the books has led to this list, which has been stewing for ages.

1. If there are, the wizarding population is so low it's amazing they haven't just been, like, bred out or inbred to death by now.
2. There are more than three countries in the world.
3. What the heck do people from any of the 200-odd other countries in the world do if they are born magical? Flounder? Get locked up? Get ignored? Run amok without anyone to teach them or stop them because there's no infrastructure? Get recruited at a young age by the government?
4. There is no way that a school full of UK kids / German kids / French kids alone (as far as we've seen) represents one third of all magical students in the world, each.
5. It would have been way easier for Voldemort to just, like, set a magical bomb and kill every last one of them, leaving the world to himself.
6. It would be like Harvard and Yale claiming they're the only schools in the world just because more people have heard of them.
7. People who are not top-of-the-line have to have somewhere to go--the equivalent of all the thousands of kids who don't get into Ivy League. Or, like, any fancy school. Ever. Are we just to believe that every single kid who has any hint of magic ever goes to one of those three schools? The average grades would be much lower, the average competency would be lower, and there would be mostly people from other countries there.
8. Or are we actually supposed to honestly thing there's only three countries in the world that produce magical children?
9. Even if there used to be dozens in every country and now there's literally only those three, there's no indication that UK, France and Germany are so out of date, or that the rest of the world has moved on to something else. Someone should have said something at least once.
10. We know that the new book is supposed to be set at least in part in the US, so there's got to be a magical presence here, too, right? Did Voldemort only have high ambitions for three schools out of all of them? Because that severely provincializes his world-threatening-ness.