In the arms race between weapons and armor the longbow was always a specialist weapon. To say that it made "crossbows look like nerf guns" is frankly untrue. The Pope attempted to ban the use of the crossbow against other Christians precisely because it was incredibly effective and anyone can use it. It had the potential to upturn society and make the wealthy man in armor vulnerable to ANYBODY. The longbow, on the other hand, was 1) primitive-- it didn't make use of mechanical advantage as effectively as re-curve bows did, 2) was something that somebody had to physically sculpt their body over the course of years to use effectively. As a matter of fact, the longbowman fed into the existing social order because he had to be tall, strong, healthy, well-fed and have as much as 10 years experience. In other words, he needed a patron, to whom he owed loyalty.
Looking at the outcomes (Agincourt, etc.) and positing the weapon on the winning side to be a super weapon is tautological. I have read the studies which Kodkod refers to and if anything they say that BOTH the French and British ideas of battle were proved wrong. The French believed that the heavy cavalry charge and the elan of knights was unstoppable. It wasn't. The British of the time (and apparently still today) believed that the longbow, fielded in sufficient numbers, would break any charge before it materialized. It didn't. But even that fails to look at the weapon in the broader social context. If you can only ever have 900 longbowmen, for example, based on what your society can support-- but you could have 5000 crossbowmen in the same society then the army you could field with crossbows is deadlier than the one you could field with longbows.
No, a longbowman did not need a patron, a longbow was cheaper to make then a crossbow and as a result more easily available. Also, before, during, and for a while after the 100 years war it was law (yes law) that all men had to train with a longbow.
For example, the Assize of Arms of 1252 law made it manditory for all lower class english male citizens to aquire a bow with arrows (did not matter if it was bought or made by the individual), and the Archery Law of 1363 made it obligitory to train with the longbow every sunday and all holidays in designated areas. As a result of that the English had plenty of longbowmen to recruit from, while most were only mediocre in the use of the longbow they could fire it, which is more then enough in massed ranks. There was no real lack of longbowmen for the English.
As a result the English at the Battle of Crecy fielded around 7,000 longbow men while the French only had 6,000 crossbowmen. Very lopsided considering that at the time France had a much larger population then the English (even then those crossbowmen were Genoese mercenaries and not French subjects).
At the later Battle of Poitiers had 2,000 longbowmen.
Battle of Nájera had 12,000 longbowmen.
Battle of Agincourt had 7,000 longbowmen (only about 1,600 other troops, almost nothing but longbowmen).
In a majority of battles the English armies consisted of more then 30% longbowmen. Not a sign that they had difficulty raising longbowmen.
In most of the battles the English won they had overwhelming archery superiority. Archery superiority at the time was much like air superiority today.
The longbow oddly enough was eventualy what lead to the defeat of England. They tried to do the same thing over and over again and eventualy the French were able to figure out how to beat it. The English got complacent and never adapted. But the Longbow was far more effective then the Genoese mercenaries even though the crossbows pack a bigger punch. Crossbow rate of fire of 1 to 2 rounds per minute is not a match for a longbows rate of fire of 3 to 6 rounds a minute.
As far as armor, 7000 arrows every 10 to 20 seconds, some of those will find a weak point and get in reguardless. Also, chainmail does not stop arrows.