Yes, high school students should be able to pick what classes they take, because high school students need to be able to take classes that prepare them for college, or for the real world. If a student plans to go to college for English, they need an entirely different set of classes than someone who wants to become a nurse right out of school.
High school students already have the ability in most school districts to choose the classes they want. They're called electives and they've existed in the American school system for several decades at least. Just like college, there are some prerequisites needed in order to graduate, so it's impossible to have a one hundred percent elective slate. But that's good training for life -- you can get your way some of the time, but not all of the time.