If I didn't think sheet music had some role in the learning process I wouldn't be presenting nearly 400 tunes here. I have my doubts about learning primarily from sheet music, but I think it can serve a very important role as a supplement to learning by ear, and the two may have complementary roles. Here are a few ideas on combining the two approaches productively. How you do it will depend on your skills with listening and reading, and on the style of learning that works best for you.
I'm sure there are other ways to use written music and listening together, but this is a start.
Here I present some ideas on combining the role of listening with the use of sheet music in learning tunes.
Learning Tunes (By Ear & Reading)