Although there are many choices, much has to do with the orientation and the expectation of the listener. Some are stopped by certain chords that are used, others accept them without reservation. It comes down to this: interpretation. Does this interpretation suit you or not? I would argue for example that Leadbelly did not know the changes for "Nobody Knows You When You're Down and Out" and therefore did not do a great job on that song. Also, when you listen to early recordings of Micheal Coleman, the renowned Sligo fiddler, the piano player that accompanied him was either drunk at the time or completely unknowledgeable about the melodic implications of the music.
OTOH, Josh White brought "House of the Rising Sun" to life with his chordal arrangements and I think even might have changed the tune a bit. The Animals picked up the tune and introduced their own chord arrangements which Joan Baez apparently used.
Tony Saletan did the chords for Micheal Row The Boat Ashore which Pete Seeger used (and I don't know if Pete ever gave him credit for that).
"All My Sorrows" by Glen Yarborough was originally "All My Trials" and Erik Darling put the Bahamian beat to it making it popular.
I advocate that choices of chords can make or break a song. Irving Berlin thought so when he hired trained musicians into his studio to run through different chord progressions for his newly minted tunes. It was, "not that chord, try another one" until Berlin got what he wanted.
Ralph Vaughan-Williams popularized "Greensleeves" with his chord progression in (the London Symphony?)
I don't know who put the Latiin beat to "Where Have All the Flowers Gone?" but Pete's original chord progression (and a rubato interpretation of the song) reflected the mood of Stenka Razin and Russian choral or balalaika folk music. The source for "Flowers" can be found in Mahail Sholokov's final pages in the novel, "Quiet Flows the Don".
Chords can make the song as clothes (the man) but if not right they can detract also.