It is an interesting question and I am one of those who plays a few covers live, but does not record them. There is another reason why I mainly play originals. Unlike some of the performers here, I do not have the musical skills to make great versions of other people's songs.
Jed Marum has described a superiority complex among some writers. Point taken, but I personally often ask myself, "Couldn't the audience be getting a bettter version of this somewhere else?" Unless I really feel that I can offer a song something which they can't get from everywhere else, I think the reson for the audience to come is that they might like my songs.
Oddly enough, I also do "Irene" sometimes. Mississppi John Hurt did a very different version to Leadbelly's, which is very charming. It is a song which you almost can't fail on. There are not many songs I feel the same way about.
All the other points are well taken. The fact is that above all it is your job to entertain the audience, rather than to try and impress them with your songwriting skills. I have no illusions that being a songwriter makes me any more creative than being a musician. I admire good musicians very much.
Busking and lots of pub gigs made me very aware of what makes material performable, so I try to put the emphasis of my gigs where I think my strengths are. If I thought I was really adding something by doing great versions of other songs, I certainly would.
Just a view from another angle...