I didn't see the show, but I've definitely heard it both ways.The American folk hymns that are minor are definitely supposed to be sung in Dorian mode, but are not printed that way. If you are a Southerner and grew up with shape note music, you just know that is how it's done, but it's not intuitive to Northerners, except for "Wondrous Love."
My organ/piano-playing aunt-in-law says some arrangements shift
to the sharp note part way through, and some don't.
I like the flat-sixth that the Aeolian scale introduces :)
The Lutheran Book of Worship uses "Jefferson" as the setting for an Advent carol, and the arranger set the key signature as Dorian, but produced a harmonization that sounds absolutely awful if performed Dorian, so he has accidentals all the way through to make it Aeolian.
Occasionally I will do a lesson with a minor key hymn, frequently "Jefferson," and try to teach sharping the sixth, but it doesn't really seem appropriate to raise the issue very often.
PGS