I think that the best answer is to assume that Hogwart's is the leading wizarding school in the British Isles, but not the only one. There would certainly be squibs in need of an education; it doesn't seem reasonable to expect them to be cast out at a young age to the Muggle world. Filtch certainly wasn't. And only a very small number of students appear to actually go to Hogwart's -- Harry's year has forty or fifty students, and that roughly gibes with the number of teachers, class sizes, length of the Sorting, etc. Which conflicts with the considerable size of the overall Wizarding population.
So perhaps like some other old, prestegious schools, Hogwarts has a legacy system. Certainly it's difficult to imagine Neville getting in based on skill or ability, or the entire Weasley family getting in based on tuition. But people from Muggle families like Hermione would be chosen for such criteria.
And so of course, even though the Purebloods who are into segregation could probably afford to set up their own school, they would infinitely prefer to have the top school in the land, and just kick everyone else out. There don't seem to be that many purely Muggle-born students, so it likely seems a realizable goal. Getting rid of mixed-ancestry students would be a long term project.
So perhaps like some other old, prestegious schools, Hogwarts has a legacy system. Certainly it's difficult to imagine Neville getting in based on skill or ability, or the entire Weasley family getting in based on tuition. But people from Muggle families like Hermione would be chosen for such criteria.
And so of course, even though the Purebloods who are into segregation could probably afford to set up their own school, they would infinitely prefer to have the top school in the land, and just kick everyone else out. There don't seem to be that many purely Muggle-born students, so it likely seems a realizable goal. Getting rid of mixed-ancestry students would be a long term project.