> They represent discovered problems in the text the author didn't foresee.
Indeed. If certain questions are asked frequently, likely the information is not presented with audience’s actual needs in mind. Content should be updated, possibly rewritten. Putting up a FAQ shows content editor washing their hands of that responsibility.
I also find it a little rude, like “Welcome to the group of people who, despite our best efforts, didn’t manage to understand what we wrote! Here are some questions you lot keep asking…” (Of course, usually it’s not that, but rather a half-hearted attempt to be helpful and/or reduce the amount of people getting in touch with questions.)
Of course, FAQs originated in places like mailing lists and usenet groups, where it's arguably a good fit, precisely because there is no real standing body of text.
The trouble is that it then got shoe-horned into being used on websites.
I think FAQs are an excellent way to present information— even information that has been previously covered— in a different way. Not everybody absorbs the same way, and not all information is presented clearly.
They represent discovered problems in the text the author didn't foresee.