Proof of mechanisms and causal relationships in biology
You are missing a much simpler and more common method. film birds in flight, then demonstrate mathematically that the wings as used are providing the lift that is enabling flight. That is direct evidence for your hypothesis. You can further test it by clipping the wings or putting the birds in a vacuum, or a hundred other strange experiments. You may want to look into biomechanics, the types of studies you are looking for are far more common. How a limb works is not really a statistical question but more of a dynamics one. 

Statistics is a secondary way of testing, used when more direct methods are not possible or terribly impractical. to use your last example yes that would be a perfectly valid way to determine a proteins function, there are limitations reasons we don't usually determine protein functions this way, not just becasue often we discover the function before isolating the protein), but even when it is you also want to test it in other ways that is true of anything in science.  This one reason science prefers multiple lines of evidence since one line can compensate for the flaws in a different line. 

"Proof by explanation" has specific connotations in biology that may not be what you are really asking about*. Often it refers to unverifiable narrative explanation which is not acceptable in science. Also you never prove anything in science, (even more so with statistics) we shy away from the term proof because it brings to mind absolute proof which cannot exist. In science you can demonstrate something or show it is the most likely known explanation but you do not "prove" it. Research is not one off events but cumulative and constructive, the more lines of evidence the firmer a concepts standing. 


What I thinks you are really asking is something much more fundamental to how science works and what empiricism and verification is. You may have better luck with this on a philosophy of science discussion, than a specific field of science since you seem more confused about how science operates. science has both experimental and descriptive research each with its own value.