The Wikipedia article on the Million Dollar Quartet recording (December 4, 1956) lists the following cuts as "Traditional":
"When The Saints Go Marchin' In", "Jesus Walked That Lonesome Valley", "I Shall Not Be Moved", "Down By the Riverside", "Blessed Jesus (Hold My Hand)", and "On the Jericho Road"
The same Wikipedia article quotes Cash's '97 autobiography:
Elvis wanted to hear songs Bill [Monroe] had written besides "Blue Moon of Kentucky", and I knew the whole repertoire.
So, I'd say that "I doubt he'd have been bothered with Appalacian [sic] stuff" is unlikely to be true. At that time, that "Appalachian stuff" was no more than a generation old. I find it extremely unlikely that he didn't know, appreciate, and (at least occasionally) play it in the same way (and for much the same reasons) that musicians today know, appreciate, and play stuff from the '70s, '80s, and '90s.'
How much he recorded of such music is another question. That music simply wasn't in his style, so would likely not have been marketable to his fans.