First, don't forget that "American Idol" is a spin-off from the British show "Pop Idol."
That speaks volumes about the genres they are looking for -- e.g., not "Broadway," not opera, not semi-classical, not rap, not heavy metal, not folk, not jazz.   They're looking for music you might hear on contemporary pop / country / r&b/ soft rock/ adult contemporary radio (which genres kind of bleed into each other).

Second, they actually have had some "folk" music performed on Idol over the various seasons. Just not a major focus.

They have covered songs by John Denver ("Take Me Home, Country Roads"), Don MacLean ("Vincent"), Dolly Parton ("Little Sparrow" and "Coat Of Many Colors") and I think a few others. As for Dylan, I wouldn't be surprised if he has refused permission for his songs to be used on the show.