I don't want to detract from the images, or the work that went into this, but, er, there are a lot of types of birds. Just among passerines (songbirds), we're talking over 6,000 species. So any sweeping statement like "birds don’t seem to bother to create a complex multi-layered harmonics pattern" is practically guaranteed to be wrong. And so it is. Lots and lots of birds sing incredibly harmonically complex songs. B…
Compare a parrot or a crow vocalizing. From that it seems some can control the amount of overtone or at least can use two different modes or something.
Seems almost like comparing humans whistling vs speaking or something (assuming whistling is closer to a pure sine tone).