Yeah, for all the nifty stuff she did, consistent worldbuilding isn't really her strong suit.

I didn't register that Beauxbatons and Durmstrang were supposed to be the only two other magic schools, I just figured they were the only ones British wizards knew much about. I mean, Italian wizards are probably not attending Beauxbatons any more than the Finns are going to Durmstrang. And I realize Rowling also isn't great at math, but if the UK (+ Ireland if we take Seamus to be Ireland-Irish not just Northern Ireland-Irish) population fills a school, sorry, but two schools for everywhere else in Europe just doesn't math.