Like Jason Xion Wang, I also thought of people like Pete Seeger, Odetta and Dave Van Ronk when I read the original post. Dave has a footnote in his autobiography about folks who belittle singers who don't write all of their own material.
I also think of folks like Billie Holiday, Frank Sinatra and Joe Williams, who did wonderful things with songs written by others, in fact sometimes the performer shows as much , if not more, creativity as/than the composer. People don't seem to complain about Frank's "cover" of Paul Anka's My Way. Leonard Cohen has said that k.d.lang's version of Hallelujah is his favorite and he now considers it "her song". Pete Seeger has said that Peter, Paul & Mary improved If I Had A Hammer in their arrangement and he sang their changes to the song.
I am not a prolific writer and could never do a whole set of material that I have written myself, but have had a couple of recordings made by others of my songs and have adopted some of the changes made by others to my songs. I am always flattered when something I have written is played by someone else.
Others have already made the obvious point that if no one sings songs that they didn't write themselves, the songs die with the composer.