"Harmonics" here isn't referring to the harmonic complexity of songs.
It's referring to harmonics as in overtones, the complexity of the overtones -- the timbre.
It's the voice of the bird, not the songs it sings.
I agree that the point TFA is suggesting seems to be about the spectral complexity at any given moment in song - what you might call the timbre - and how nearly pure the tones in their spectrograms are. (They don't specify what species they're showing. Looks like several, but I can't ID them by eye.)
My point is that no, in fact, many bird species produce vocalizations that are indeed spectrally complex (beyond even just harmonic stacks) from moment-to-moment.
Take a look at song from a blue jay (https://www.remoteenvironmentalassessmentlaboratory.com/expl... ; not the best example, or one I produced, but an easy one to hand), particularly the syllables near the end of the clip. That's an example of a complex timbre.
And lots of species produce song and calls with features like this.