From a music theory standpoint, I'm not sure it's all that surprising that some sad songs are set in a major key. In any major key, only the I, IV, and V chords are major chords. The II, III, and VI are minor, and the VII is diminished (which sounds even sadder than a minor chord).
Let's look at Yesterday, as an example. I don't have the song in front of me, so this is from memory (which can be shaky sometimes) but I'll give it a go anyway.
The first chord of the song if F major. However, if I remember correctly, the song quickly tonicizes D minor, going with E minor/A major/D minor.
If I had the chord chart, I could explain it further. I'm pretty sure that the melody doesn't even stick to the key of F major very long; I think it leaves it where the lyric says "All my troubles seemed so far away," which occupy the second and third bars of the song.
I guess what I'm saying is that a song can spend a great deal of time harmonized by minor chords, even though the original key signature is major.
"We are not provided with wisdom, we must discover it for ourselves, after a journey through the wilderness which no one else can take for us, an effort which no one can spare us, for our wisdom is the point of view from which we come at last to regard the world." --M. Proust