Can the harmonic series explain the origin of the major scale?
Emphatically yes, but that doesn't mean the major scale is literally contained in the harmonic series. We seem to be dealing basically with a black-and-white fallacy here: there aren't just two options “the entire major scale is contained in a single harmonic series” and “the major scale can't explain the major scale”. Rather, the scale is constructed from multiple building blocks that use the harmonic series.
As you said, the major triad is so well supported by the harmonic series that this can hardly be controversial. That's basically enough to explain the entire scale too if you just make use of the obvious candidates for a chord sequence: the Ⅰ chord gives the Ⅰ, ⅲ and Ⅴ scale degrees as 4:5:6 ratio, then the chord which has the original Ⅰ as its fifth is the Ⅳ subdominant, which adds the Ⅳ and ⅵ degrees, and finally the major chord on the Ⅴ degree gives you also the ⅱ and ⅶ.
The scale thus derived, using only the first five overtones in the harmonic series (plus extra octaves), is the Ptolemaic diatonic major scale. It could be argued that this is not the historically most relevant one, with the only 3-limit Pythagorean being an alternative. The fact that Pythagorean melody and Ptolemaic harmony can be pretty well approximated by a single tuning system – that's a meantone temperament, of which 12-edo is but one amongst many – is perhaps the centralmost feature of Western harmony.
Why use those particular three chords, Ⅰ, Ⅳ and Ⅴ to derive everything? Well, they are the obvious candidates insofar as you need to move the fundamental only by the simplest ratio – the 3:2 fifth. It is certainly possible to move around major chord by other steps, in particular by the mediant. That gives the ♭Ⅲ or ♭Ⅵ chords, which indeed are quite common in bluesy rock music and sound very tonal. Those songs sure enough tend to use not so much the full Ionian diatonic scale as the more flexible “blues scales” for their melodic content. It so happens that Common-practice music has not gone that road, which probably is mostly because those kinds of chord changes are much more natural for a parallel movement, sliding chords around the guitar neck style which CP so avoids. But I'd say that both approaches are clearly based on the harmonic series.
Similar story for
But the next strongest thing is a diminished triad
– kinda, however 7:6 is significantly narrower than a 6:5 minor third or even a 12-edo minor third. So it's no surprise if a diminished chord sounds dissonant although 5:6:7 are low integer ratios: because it would be way out of tune from that interpretation. A true 4:5:6:7 chord however sounds pretty amazing. Why does the seventh harmonic not feature in standard Western scales? Well, Western music has been to a huge degree shaped by the composers of the Baroque who loved symmetry and modulation. The playfield set up this way from only 5-limit primitives allowed for and enourmous richness that was carved out in the 18th and 19th centuries. Now, if you toss 7-limit intervals into that kind of music, everything would become a heck lot more complicated. 7- or even 11- and 13-limit intervals do feature in other musical traditions, but the Western composers just happen to take the more symmetrical road of 5-limit meantone music.
I do think extending Western counterpoint with 7- and 11 limit intervals is a great way in which music could evolve in the future. Just it's really quite difficult to keep the overview over the fine-grained scales and still keep it melodically graspable, about the only composer who has really been successful in that direction is Ben Johnston. Better computer frameworks, if not AI, might soon help as an assistant.