Margarita,
interesting points you make, could be a thread of its own, but I respond here. In Germany, it was a well accepted pastime for 'serious' composers to write folksy melodies or to write a new melody to a known song or poem. You can find in German songbooks notes like this: "The tune we publish for this song is the one most often sung though there are more than 100 tunes known for this song". You can find many known composer names (Brahms, Mozart etc.) with folksongs, i.e. songs that are sung to this melody by the people. Some songs from operas, like e.g. "Der Vogelfänger" from Mozart or at least one song from "Der Freischütz" by Weber have become sp widely known that they might now be listed in folksong collections.
Schubert is a different story, for his Lieder are much too artful to be sung by normal humans. I do not think that any of his songs is sung today using his melody.

Wolfgang