I wouldn’t worry as much about going to an expensive school, but instead about instilling a great work ethic.
It’s all about the scholarships, at least that’s how my parents did it for me: mediocre schools, no extra tuitions, just asking lots of questions and getting good. Soon got scholarships and got a great education for a relatively small amount of money.
More expensive schools are so because they spoon-feed students. Once they get into university, they burn out because there is no one to spoon-feed them. Same once they get a real job. So you’ve got to make sure you’re setting them up well.