I would think that whoever put together "I Can't Help Falling In Love With You" had heard the old French song " Plaisir D'Amour", which was covered by Joan Baez and a few others over the years.

"Some wise-add said
'Only Russians are fools'...."

Yeah, THAT one!

And "It's Now Or Never", of course, was "O Sole Mio".

I have heard some of Elvis' slow numbers sing as lullabies to put babies to sleep that they surely could be counted as Folk Songs by now. I have felt for a long time that any song sing as a lullabye is being used in a Folk context, so it qualifies. Anybody whose infant won't go to sleep even after you have sing every official one, and has to resort to any slow song you can think of it had had to start improvising further verses to whatever you were singing

" When you wish upon the sun,
Makes no difference what you've done.... "

Earth....what you're worth
Sky....No need to wonder why

....and on into the night. That's the folk process in action. Writing them like and how they usta back when. Putting kids to sleep. Hoeing the garden etc. It's the use you put the song to.....