The British spent years preparing for war. The motto was “six feathers from every goose”, to the fletcher for arrows. Every goose, for years in a country with millions of inhabitants. The Battle of Agincourt was not a casual encounter, both sides spent half a generation in preparation. There were 5,000 longbow men. That works out to about 200 arrows per bow. If you didn’t have enough arrows for them, why even bring them along.
Longbow men did act as light infantry, but generally only those sent through the wood to attack the French flanks, who when they ran out of all the arrows they could carry, would pick up abandoned arms - the battlefield was full of them - and make harassing attacks on the French flanks.
I've done that!
(Well, she was part Japanese also. But it sounded familiar...;)