I agree, though, Hofstadter and Dennett have done an extraordinary job of devising such an answer. There is lots more work to be done, of course.
But what if the answer to the "weirdness" question is, in fact, that it's just an ill-formed question? What if it's just an illusion, and your brain is tricked into seeing something magical about itself where there is nothing there?
Apart from a mystical explanation, I cannot imagine any satisfying answer to the question (i.e. one that doesn't leave you feeling uneasy like Hofstadter's answer does to most people), which usually means that there's something wrong with what we're asking, not with how we're trying to answer it.
I really don't think the question is ill-formed, though. There is a big explanatory gap. Denying that is simply dishonest. It would be like, if you don't know why a bicycle is easier to balance when it's moving, pretending that there is nothing to explain. "Oh, it's just bicycle parts."