maybe just blindfold the bird, instead of removing the eyes
This sentence makes me wonder: are they docile because it genuinely calms them, or are they entering a sort of helpless automatic panic mode, like "OMFG no escape, can't fly, can't move"?
The latter sounds like the bird is getting a terrifying experience.
Looks like the humane way to test, according to the article cited above, is to affix magnets to the eyes/beak to see if it disrupts anything. In any case, in retrospect, I personally feel my suggestion to remove the beak is more cruel than removing the eyes. Both are hideous to contemplate. But, a bird without a beak? That's just a miserable thing to imagine.