First of all, wow! That is a great song and great performance. Wish I could sing like that.
Second, I would say the entire song is an elaboration of a half-diminished chord (B, D, F, A) with a kind of contrast to an implied A minor chord near the end of the refrain.
Rather than avoiding the C it provides the flat 2nd degree.
The degree that is avoided entirely is the G.
I feel the avoidance of G and use of A between the B and F makes the A seem like a chord tone. That point could be debated, because the A is always on a weak beat except for the end of the refrain.
About that A for the end of the refrain. It goes from E to A back to E so that seems like an implied A minor chord. I'll call this a harmonic contrast rather than a cadence, because technically you can't cadence to a diminished chord. Funny thing is, it feels like a cadence to me, even though it doesn't fit the textbook definition. Some people call root movement by whole step (like bVII to i) a cadence in folk music. Here the root moves a whole step from A to B and so is similar.
Personally I love the half-diminished sound. So it doesn't surprise me that this melody is so haunting and beautiful.
Just for fun I set the tune to some chords. To deal with the "problem" of the tonic chord being a diminished chord, I simply avoided the issue and used a G chord so the B becomes the third of the chord. I suppose this obscures the locrian quality of the melody. But I did get some nice root progressions by 4th/5th, a relative major/minor, and a bVII all which seems to me appropriate for folk music.
...I find it hard to see the phrasing on that score, so in ascii...
|G |dm |em F |G | |
|G |dm G |em |G | |
|em |F G |
|em dm|em F |
|em |F G |
|G |dm am|am |dm |G |