I teach tunes for 2 reasons. Some tunes are carefully curated to contain examples of techniques the student should be learning. The targeted techniques are: good tone, good intonation, good rhythm, and personality. The other type of tunes are standard fiddle player vocabulary. These are tunes that you should know to join a jam session or play with other acoustic musicians and call yourself a fiddler.
What to do with that technique
If you only play songs that someone else has supplied to you, you are more of a technician and less of a musician. The very word "fiddle" promotes the idea of customizing or changing something- in this case notes! I would consider it a failure as a teacher if my students sound exactly like me. I introduce musical theory concepts that allow the student to begin customizing existing songs and creating new ones, expressing themselves as individuals. It's important for each musician to have their own sound and musical ideas as an artist. This allows the student to develop the ability to play with others in many settings-either informally in jam sessions or as a professional musician performing, writing and recording.
Tablature Versus Notation
I believe it is vitally important to train the ear, even to the exclusion of the eyes. To that end, I use tablature and recordings to convey musical information, with the goal of training the student to eventually be able to play what they hear without visual aids or outside instruction. i.e. no sheet music! Notation gives all the information through your eyes. Recordings get the tune in your head, tablature gets it from your head to your fingers with the minimum written information. This forces the student to rely on their ears primarily. Ideally, tablature is used temporarily while the student learns the tune, and discarded as soon as possible. This prevents the student from getting fixated on one version of the song and creates opportunities for improvisation and personalization.