Where does the major scale come from?

When I’ve tried to look into where the selection of notes for the major scale originates, most of the answers either focus on history, or say something about the choices of notes for this scale being derived from the harmonic series (the series of overtones). Here are the first several notes in this series:

If we start on low C, the next harmonic (double the frequency) is the C an octave higher—middle C. After that, the note with three times the frequency of low C is middle G, introducing a perfect fifth. (The ratio of any G to the C right below it is also fairly simple, at 3:2).

The note at 4x the frequency of our bass note is two octaves higher—in this case high C. Then, we introduce the third (E), and see the fifth again (G).

Given this, the harmonic series is great as a way to explain where major chords come from—the perfect fifth and major third are both found in relatively low harmonics. However, after that things quickly fall apart. The next new note introduced—the B flat—isn’t very close to an actual B flat since it’s 31 cents flatter than it should be. Also, B flat isn’t part of the C major scale at all.

If we extend our search outwards harmonically, we can use a few major chords to build a major scale. We have to look at the major chord a fifth above our base note (in this case G major) and the chord a fifth below (F).

This gives us all the notes of the C major scale, and no others:

It’s kind of interesting that the tonic and the fifth are the only notes that come from multiple of these chords. However, I find it kind of unsatisfying that we have to look in two places—both the harmonic series, and the nearby fifths—to build a major scale versus there being one source that accounts for everything.

Maybe someone with better music theory knowledge has a tidier explanation of how the major scale comes about, but I haven’t run across one in my (fairly brief) searching.