Bill D wrote: It feels to me like the musical need to 'resolve a chord'.... at least in Western music. That 3rd note allows a sound to be 'complete'. In poetry and song, there is a similar almost implicit structural need.
   Those who know music theory may be able to say more about it.

It's not really that the third resolves a chord -- a modal chord consisting of the first and the fifth (do and sol, so e.g. C and G in the key of G) doesn't sound as if it is particularly lacking. Indeed, modal chords often sounds particularly full. (This is why DADGAD tuning sounds so rich, e.g. -- the D chord you play in DADGAD is DADAAD, which sounds exceptionally strong.)

What the third does is define whether the chord is major or minor. A modal chord, with just the first and the fifth (C and G in the key of C), can be either. It is only when you a the third -- a major third or a minor third -- that you know whether the chord is major (i.e. Ionian or Mixolydian) or minor (i.e. Aeolian or Dorian). So adding the third to the chord defines the scale of that song.

However, that's not really using the third as a "third"; it's the second of three notes making up the chord. You don't use the first, second, and third of a scale (i.e. C, D, E in the key of C) in a chord.