There are claims that carpenter axe was very effective against even tanks, see here:
translate.google.com
translate.google.com
Second one exactly confirmed from German side. But of course this doesn't mean that axes were devastating on the fields of WWII or mortars are good in anti-aircraft defense (was a confirmed case too). Just good use of place and time. And courage with luck, of course.
Jokes aside, you said about certain battles. Then why are they only few, if those longbows were SO great? Answer seems quite easy - enemy quickly adapted to the new danger.
P. S. I remember the Japanese aces in the beginning of WWII. The were real elite and were superior as pilots over US's. But the US adapted and started to produce flying logs with 2 kilohorse engines and shot the aces from the high. And Japan could not replace them so fast, then they were defeated in air warware. It is very simplified explanation, but what I wanted to say, that war is not which unit better, but whose system is better.