In a sense, Harry Potter is the worst when it comes to class divide.
What let the kids go to that boarding school is the innate ability to do magic. A privilege you are born with, and that is dependent on your ancestry. Even if "blood purity" is a recurring theme for the bad guys in the books, it is made clear that the ability to do magic is like a gene, and if you are born a muggle, you will stay a muggle.
The reason it is worse than selection based on wealth or even nobility is that while it may be something you are born with, this is also something you can acquire, not so much with magic.
Not only that but there is essentially no drawback to being a wizard. Wizards tend to dislike muggle tech, but there is effectively nothing preventing their use, it is just that they can do better with magic. In the same way that being born rich comes with a lot of advantages and very few disadvantages.
The idea of innate magic ability that make those who have it strictly superior is extremely common, and honestly, it works, but if you want a story where the idea is that anyone have their chance, which is how I interpret the idea of "not just wealthy kids", then Harry Potter is not that.
That being said, as far as I'm concerned, it doesn't take anything away from the story, not everything has to be a political statement, and in fact, making a political statement often makes the story worse.
Lots of fantasy settings end up supporting some really bad worldviews if you look too far into them. I highly doubt the creators of Avatar: The Last Airbender realized they were creating a society of ethnostates enforced by an all powerful pseudo-immortal autocrat supported by a secret society. They just wanted a reason for the main character to go on an interesting journey to save the world.
It became a problem for me when they tried to make a sequel that took the setting more seriously. Maybe one person chosen by lottery at birth shouldn't be allowed to kill heads of state without any oversight? The villains who were opposed to the chosen one's uncontested rule ended up making more sense than the enforcers of the status-quo.
There's a funny video explaining why the Disney direct-to-tv movie Sky High is actually propaganda for fascist eugenics. https://www.youtube.com/watch?v=iIdbLUm-ez8 The "some people are just born superior" out is an easy excuse when you're trying to come up with a reason why the main character and their cohort of whacky friends have the ability to save the world while everyone else is helpless, but taken to the logical extreme is blatantly fascist.
