Mae here.
Actually, we DID do exactly that to control for the possibility that they were simply responding to the louder noise. During playbacks we also played a series of neutral natural sounds (rosella bell calls) that we matched in amplitude to the volume of the alarmed whistle, the birds paid absolutely no attention to them. We also played whistles recorded under non-alarmed conditions at 'alarmed' volume, again the birds did not respond. They only responded to whistles recorded under alarmed conditions, played back at alarmed volume (and once or twice to whistles recorded under alarmed conditions but softened to the volume of non-alarmed whistles). Give us some credit, a lot of time and thought is put into experimental design.
In answer to the previous note, indeed the 'alarm' is linked with 'getting out of dodge'. As such, we can not distinguish yet if the whistle is an intentional 'signal' or simply a 'cue', we state this in our paper. Possibly the sound is intended for another purpose (mate choice? species recognition?) and the fact that it also indicates alarm is a side effect. Irregardless of intended function from the perspective of the 'signaller', it still produces a different sound in alarmed flight and co-specific birds are still using it as an indication to flee, and benefiting from doing so.
Another note, It is different from the wing noises of other birds in that the upbeat and downbeat produce very different tones resulting in a 'pulsed' sound during flapping. In other birds you just hear an indistinct 'whoosh' during flight, in Crested Pigeons the pulse enables you to hear the faster alarmed flap rate quite clearly. If all birds flew away every time they heard another bird fly (without knowing how alarmed that bird was) they would never get anything done! I expect many species deliver information through noisy take-off's in alarm, its just that the Crested has exaggerated it to communicate the degree of alarm more reliably, thus negating the need for additional vocal alarms.
It is actually quite a neat system, it is more inherently honest/stable then standard vocal alarms in that there is a much higher price to 'faking' an alarm, as the bird MUST fly away, and hard, in order to produce an 'alarmed' type whistle. Thus eliminating any benefit it might have received through getting unlimited access to resources. In vocal alarm systems, there are practically no costs to producing false alarms. Additionally, the bird can't 'forget' to produce the sound, the whistle is inherently graded and it also does not take any extra effort to produce. Additionally, since it causes all the other birds to also flee, it offers a benefit to alarm signalling, the escape of the individual who first flies away will be 'hidden' in the flights of others and the huge noise produced by the escaping flock is also likely to double as a way to scare/distract the predator. All at no extra cost, given the bird would be escaping anyway.
Anyway, sorry you didn't find it interesting, each to their own. I know i loved working on it, and i think they are fantastic little birds, lots of character :)
p.s. Sorry for the super long post. Ooops! Overenthusiasm.