(Also, songs are often written in one key, then transposed in the studio.)
I'm not sure why F would be more popular than D and A. One hypothesis would be that it has fewer accidentals; it's not a fun key for guitars, but many contemporary pop songs aren't guitar-driven anyway.
EDIT: Thinking about it, E minor is fairly common in phrygian scale (it's the tonic chord), however, phrygian, unlike dorian or myxolidian, which are used at least sometimes, is super rare -- its dominant chord, B dim, is dissonant, which kind of sucks.
Functional harmony kind of goes out the window once you stray from the major/minor modes anyway. You have to lean heavily on the root to maintain the tonality, or else the perceived key center tends to shift.
I think it's safe to say the effect of metal songs using phrygian is negligible.
By that measure it's only natural that Em (iii) is more common than E (which is non-diatonic to C).
Emin vs Emaj - my gut would say Emin is going to fit in with keyboard/piano stuff easier, and there's more of that type music overall. When I think Emaj, I think guitar stuff (EAD stuff), and there's been less of that overall compared with keyboard-based music.
It’s useful for dreamy and somewhat static chord progressions though.
D/E/A - again, ime - were always "guitarish" songs. GCF always seemed to be more "pianoish". Just my own observation/experience - it may not be true of everything, but seems to be true about much of the stuff I've listened to (and tried to play).
A major (A/D/E) is probably the easiest key to play. G major is arguably the most useful, because you have access to all the important chords in open position: G, Am, C, D, Em.
Also, in this way we can say "C major" or "C minor" to refer to the key of the song. We use "natural" when its important to distinguish from a note that has a sharp or flat - e.g. C# versus C natural.
In Italian, how would you refer to the key of Db major - "Db natural"? How would you refer to the key of Db minor?