Well stylistically speaking....Radiohead really doesn't have a whole lot in common with what Jack has done in any of his bands. "Genius" isn't a word that I really like to use in regards to music and I wouldn't go as far as to say use it to describe Jack (or Radiohead), but I think what a lot of people like about The White Stripes is that they're a newer band that swims in the classic rock stream but still appeals to indie fans, punk fans, and young kids who don't know anything about music. I think their wider appeal is what makes some people think he's a "genius".
As a songwriter and a lyricist he's pretty good (but let's not over exaggerate). As a singer he's unique sounding and it works with his raw, garage-y music but still isn't outstanding. I think what he does the best though (other than balance a million things at once) is play guitar because is soloing is a hair away from being out of control but he stays rooted by playing the rhythm parts in the middle of them and by having a steady drummer like Meg to keep the beat.
Or at least that how I see it.