Most of them have been covered. There is an obscure one called" Beautiful Lake Anconi",
a fiddle tune from Cape Breton by a Laurie ( ? )

One of the most simple plaintive tunes to me is "Little Birdie".

Pete's songs would qualify. Where have" All the Flowers Gone" and" Turn Turn", "Bells of Rhmyney".

"Come All You Fair and Tender Ladies",    "Pretty Saro", "Wagoner's Lad"

One of the great early popular songs written which is a jazz standard is Jerome Kern's
"All The Things You Are".

Thomas Moore. "Believe Me If All Those Endearing Young Charms"

"Tiurse Moi Chroi" from Ireland

Texas Gladden's version of "I Never Will Marry".

"The Foggy Dew" from Ireland

"Been In The Storm So Long" (sung during the Civil Rights Movement)

"Summertime" George Gershwin

Cole Porter could write a melody.

For a simple, clear heart-felt tune "Dink's Song" collected by Lomax.