Does the bird study attempt to rule out the explanation that people are using the number of birds claimed to be saved as a cue to how many birds there are in total and so adjusting their estimate of how rare they are?

I place little to no value on the lives of individual birds but I do value the existence of some species of birds (because they are aesthetically pleasing) and I value biodiversity for both aesthetic and practical reasons. I therefore place some value on maintaining a viable breeding population of particular bird species but the marginal value of additional birds over this level is very low.

If someone asks how much I am willing to pay to save a small number of birds this is evidence that the birds are rare - the fact that someone considers it worth bringing the issue to my attention for a small number of birds suggests as much. If someone asks how much I am willing to pay to save a very large number of birds this is evidence that the birds are not rare. I would pay more to save the last 10,000 of a rare species from extinction than to save 1 billion from a population of several billion.