I contend that if willow water works in any way at all, it would be because of the presence salicylic acid, rather than any auxins. Auxins that are steeped from willow would be at an incredibly tiny concentration. Of course, only a tiny amount is needed, but just think about the tiny amount that would be inside the cells, and then only a fraction of that gets steeped into water that is a million times and more the previous volume that was inside those compromised cells. Where could such an enormous dilution rate be valuable?
But the salicylic acid is what is absent in other applications and I could see it that might make a difference. There are scientific papers that document its effect on plant cells. None that I know of that connect to any rooting results, but a certain kind of cell stimulation is real, and that might have a roundabout effect. I wish I could be more specific, but that was so long ago that I looked into this.
The whole question of willow water usage is often brought up in forums, etc, and I always offer the same challenge:
Don't show me anecdotal info or claims, show me factual data with even a modicum of science in it. I never get any replies.
All this said, I would never completely dismiss the subject, unless I see science disproving it also.