Are chords important?

Chords provide harmony, and usually rhythm, that both support the melody and also contribute their own dimension to music. Chords normally express the feeling of the music and the changes in feeling that form the overall arc of a piece of music.  Certain chords are usually pleasing, disturbing, happy, or sad, but their feelings also depend on their context; one chord can sound very different in different chord progressions.  And a melody can sound very different depending on what chords it is played over; so chords really do contribute their own effect to the sound of a song.

Using songs from The Beatles as examples, the song “Here Comes the Sun” is happy.  It uses only major chords.  It can be heard in this video:

By contrast, a sad song by The Beatles is “Yesterday.”  It uses both major and minor chords, with the minor chords placed to emphasize the sadness of the story told in the song. Here is an early video of Paul McCartney (of The Beatles) singing and playing it solo: