Random discovery from attempting to find songs from game OSTs for my alarm, which resulted in searching my music library for anything with "ost" anywhere in the track info:
P!nk's "But We Lost It" and Radiohead's "At My Most Beautiful" are piano-heavy songs that open with a prominent middle C.
They're in very different contexts (my best attempt at theory follows), and go in very directions, but they start with the C and go from there. Which I guess you would expect 1/12 of songs to do, but I bet there's some bias.
I wonder what the distribution of opening notes is? Does it change if isolated to songs that are piano-forward?
attempts at piano theory
So my best attempt at describing how they're in different contexts:
P!nk opens with the C as the sixth ("submediant", says DDG) over an E flat major in second inversion, and alternates with a B flat over the same chord, then repeats that melody over a B flat major in root position (with some other stuff). Vocals start with B flat at then fall into a C-B♭ cadence mirroring the opening notes (they say EVery THING is = B♭ B♭ C B♭ C B♭)
In Radiohead it's the opening to a descending 4th C4-G3-A3-C4 melody, with the C as the dominant (fifth) of the first-inversion F chord above it, and then the third ("mediant" but does anyone really call it that?) of a root-position A minor above, and then it gets all fun and dissonant.
Music is wild! So many combinations of sounds, so many ways for notes to relate to other notes. I know enough jargon to describe things, but have very foggy ideas of the theory around how to use those relationships.