I guess my question is this: if you say you're only going to teach voluntary, interest-based things, how do you make things interesting? How do you make math interesting to a kid who doesn't care if he knows how to add? How do you make history interesting to a kid who thinks it's awful?
There's a lot of value to a well-rounded education. If you don't "force" kids to explore a variety of areas, they'll have a tendency to become focused and centered on the one thing that really gets them off. That's great, unless they start blowing off other subjects and whatnot. It's the reason most universities require a variety of general education classes for all freshman.
So I think we have to "force" everyone to study different topics, not just the ones that intrinsically interest them. The key here is "forcing" the kids to like the material. How do we do this?
Well, I don't have a great answer to that. The only thing I can point to is that one great teacher we've all had (or I hope you've had). That one teacher who was able to just turn you on completely. But it was the teacher that was special, not the material, not the procedure, not the tests, not the class.
My favorite example is Mrs. Sams, my calculus teacher. When we got to the fundamental theorem of integral calculus, we held a mock wedding between the derivative and the integral.