For a really outstanding exponent of this style, see HELENA TRIPLETT. If you don't believe me, ask Dick Greenhaus.
The secular and religious styles from the Southern Appalachians are probably very close. That does not necessarily mean that the secular style "arises from" religious singing. My understanding is that Southern Applachian music is basically rooted in Scotland "Scotch-Irish." Bronson, on analysing a few hundred old tunes and comparing English, Scottish and American tunes on the basis of mode, etc. concluded that the American tunes were Scottish than the Scottish ones (presumably because more isolated from outside influence).
It seems to me that the old singing style does not "wander around" the tune, but ornaments it. One notices in particular a lot of high grace notes, especially at the end of a line. Ornamentation if done just right can add a lot of power to a song, but it is extremely hard to do just right. Another advantage to a capella singing is that it lends itself to extremely subtle use of rythm, which is impossible when accompanied by a chinky-chinky guitar. Some Appalachian signers make extremely good use of these rythmic possibilities. (And some guys I know just ignore the fact that a song has a rythm at all.)
Irish singers (off the top of my head, Joe Heany and Sean Mac Donnchadha) also have ways of ornamenting an a capella tune, but the kind ornamentation is quite different. And Irish a capella singing also has a kind of rythmic ornamentation that seems unique (and which gets annoying if used with non-Irish songs). To me also, West Virginia singers sound quite different from North Carolina singers, but maybe that's because my sample is so small.