For any learning, one can’t do advanced courses in it without first knowing the ABC’s. You can’t write a word, paragraph, or essay without knowing the basics of how to put it all together.
The same goes in high school. It’s good for our country when essentially everyone has been exposed to the basics of life (Bio/Health/Chem), direct problem solving (higher level Math/Physics/Personal Finance), philosophical problem solving (English), and of course, one is doomed to repeat History if they don’t know about it (in depth). Add in languages to communicate with more of the world and you’re set. 
ps More classes can fit into the parentheses above, but going without any of them has students missing a ton of their adult ABCs. Our country and world would not be a better place. Self-directed sounds great, but how can one know to learn about things they don’t know exist? And just because a student doesn’t like a type of classes doesn’t mean they should remain clueless about it for the rest of their lives.