I selected 60 songs that have stood the test of time - tunes such as Tea for Two and I Can't Give You Anything But Love and It Had To Be You. I then noted the chord with which they start. I am referring to the first chord of the first bar of the Chorus (i.e., omitting any anacrusis).
But this leaves twelve tunes (20% of all I studied) that begin on the major chord of the home key but are more subtle and complex, perhaps with elements of sadness, nostalgia or melancholy. These include such tunes as I'm In The Mood For Love, Smoke Gets In Your Eyes and I'm Getting Sentimental Over You. If you look at the inner workings of these tunes you find minor chords, diminished chords and other surprises (such as a 7th based on the flattened third note of the scale in I'm In The Mood For Love). These chords make the tunes harder to learn but they also give the songs their distinctive colours and make them linger in our minds, it seems to me.