""Why do I think I have free will?" "Because I do," is a perfectly good answer (assuming free will)."

Disagree. Notice how this answer is only 'good' assuming free will. But our assumption of free will is exactly what we are seeking to understand the cause of. We can assume free will is correct and that this is adequate to justify our answer ('because I do'), but then we have only re-posited the assumption in the consequent.