You rarely solve hard problems in a flash of insight. It's more typically a slow, careful process of exploring a branching tree of possibilities. You must pause, backtrack, and weigh every alternative. You can't fully do this in your head, because your working memory is too limited. Writing is the external medium that affords the time and precision necessary. Serious thinking must be done in writing. And that's why you can't outsource your writing, because then you're outsourcing your thinking.