[crazygringo]

I think the explanation is even simpler.

For a loud, high-pitched sound produced from a tiny object like a bird, is it even physically possible to have the loud resonances needed for overtones and timbre? I think it's just physics.

Whales are "signaling" over huge distances too but have plenty of overtones, since that's what low frequencies in huge cavities produce.

I'm not sure the dichotomy between "aesthetics" and "signaling" you suggest actually exists. A peacock's plumage is certainly aesthetics but also certainly signaling. And what is a birdsong's melody if not aesthetics?