It is quite common knowledge that longbows most likely did not penetrate the plate armor worn by the French chevaliers at Poitiers and Agincourt. However, how effective were these longbows in hampering the fighting ability of the French foot knights when they reached the English lines? Indeed at Poitiers Peter Hoskins in In the Steps of the Black Prince insinuates that the battle was won by Anglo-Gascon discipline, NCO experience and French incompetence but not the English archers. However, did the longbow archers play a big part by disrupting their foe's formations, wounding soldiers and making it harder to swing with arrows lodged in your armor?
Now as a caveat I must warn you that the question you are asking is pretty specific but I'm going to give you a general idea of the efficacy of French foot soldiers against the English forces.
Firstly, what you say is generally correct. French armour at the time was extremely well manufactured and using slopes and various inclinations in the armour, arrows were generally ineffective at full penetration of steel plate even at short distances. With that said, because high quality steel was expensive many soldiers especially footsoldiers, who were generally lower in rank then their mounted counterparts, could not afford full steel plate armour of the grade necessary to prevent arrow penetration. In this case, wrought iron armour or armour mix and matched from different craftsmen was used which a) decreased the overall effectiveness of the armour and b) caused certain parts of the armour, especially limb armour, to be relatively much weaker than breastplate armour. In this case it would be safe to say that while the French had superior armour, the prohibitive cost allowed English longbowmen to still remain effective at least on ground troops.
With that, running in 50 - 80 pounds of plate armour made footmen easy targets. Fatigue, heat exhaustion, and time wasted stepping over fallen comrades also hampered the effectiveness of French ground troops, giving longbowmen time to fire at the French line. In addition, like you said, I can't imagine those who did make it with a chest full of arrows being particularly effective at swinging a weapon.
As per the rest of the French army, English longbowsmen trained their arrows on the horses of charging knights. Because horses were generally less armoured, crippling a horse and throwing the rider was an effective way of eliminating mounted knights' combat effectiveness. In this case, the longbowmen excelled, but at an effective range of about 220 - 300 yards, a horse in full gallop could cross that in under a minute giving the archer about a dozen shots before the enemy was upon them. Furthermore, equipped with better armour, I'm unsure if the longbow was effective in crippling the mounted knights of the French.
In short, I theorize that longbowmen were quite effective in disrupting the attack strategies of French footmen. But as per the horse back riders, without more data it's hard to say.
As an aside: This video on youtube describes a process where arrows were shot at armour and effectively stopped by the plate. Some of the comments are worth reading too.
I don't know if this is applicable but I hope it can guide further research.
There have been many attempts at experiments over the last couple of decades as "experimental history" has gained respectability, but, it's very hard to reproduce the past in detail.
For example, in "Firepower: Weapon's Effectiveness on the Battlefield 1630-1850" the author Hughes notes that based on contemporary test of accuracy, impact and rate of fire of volleyed muskets, the historical flintlock would have 100 yards have swept the battlefield as effectively as a modern machine gun. Yet, clearly that did not happen. The reason that modern black powder muzzle loaders are so deadly lays in the quality of the manufacturing of the powder and weapons, borderline magical by historical standards, especially the powder, the amount of training of people using the weapons in test and, oh yes, the inability to measure the effects of people shooting at you on accuracy, rate-of-fire etc.
Likewise, it's impossible for modern researchers to replicate the effects of the historical longbow. We don't know for certain how the longbows were constructed, methods were usually craft secrets or handed down through families. We don't know how representative museum samples are or how time may have altered their composition. Likewise, it's difficult for use to replicate the low and varying quality of the metal used in armor and the arrows. Most of all, no living recreator, no matter how dedicated can duplicate the nuances of warriors trained from the age of five to shoot bows or fight as armored knights. Niether can we accurately evaluate the effects of actual combat of the effectiveness of the warriors and their weapons.
Still, based on modern experiments I would hazard that the longbow weren't really that lethal but instead had two primary effects on the control of the battle field and the ability of the French to advance.
Firstly, the longbow did allow the English to sweep from the field or prevent the taking to the field of any of the unarmored French units. The Genoa crossbowmen never got close enough, none of the French peasant infantry, nor their era's version of light calvary could stand the archers. The French knights were left like modern tanks attacking without infantry support.
Secondly, a longbow arrow didn't have to kill, wound or even penetrate the armor to become an important factor on the battle field. It wasn't just a matter of arrows either growing deep into flesh or bouncing off. The majority, or at least a large number of the arrows, would have penetrated slightly into the armor, and then lodged there. Period descriptions of the French dead definitely mention that the bodies seem festooned with arrows.
However, it's likely that the arrows functioned like the Roman Pileum spear, immobilizing the opponents armor rather than killing outright. As the French knights lumbers across the mud, they faces sleet of arrows and pretty soon every night would look like pin cushion with three-foot/one-meter yew dowel sticking off in every direction. Since the knights were marching shoulder to shoulder, in mud, the entangling and tripping effect of all those rods must have been enormous.
In this scenario, the arrows served more to velcro the knights into one hapless mass in which individuals could not move forward, backward or regain their footing. The French lines made it to the English stake line and I do imagine those on the front got shot through or speared. Then the rest would tangled by tangling arrows, mud and the odd dead guy. None of the rest of their force could come to their aid in the face of the longbows.
Then the English just darted out in the big tangled pile of French and gave them the old boot-dagger-under-plate coup de grace. Certainly, IIRC period sources don't seem to explicitly state the French were killed by the arrows. The did record many cases French smothered in mud although whether they recorded those deaths because the smothering was major cause of loss or because of the horror such a death provoked at the time, we can't say.
Poitiers and Agincourt, like all battles of annihilation, were so lopsided because one side did almost everything right and the other side did everything wrong.
Clearly, the longbows were just one element in a highly integrated battle plan that had deep roots in English doctrine and tradition at the time. English rule of law, firm protection for private property and relatively larger and broader middle-class or lower nobility, allowed the longbow to become a major battlefield weapon and to be available for Henry to anchor his strategy on it.
Conversely, the French's aristocracies sneering contempt for their lessors, their impoverishment of the people and their weak rule of law meant they had no substantial, effective and trustworthy talent pool to draw on. Instead they had, armored knights, hired mercenaries and peasants for road bumps. The French could not have replicated the English tactics with radical long term change to their social order.
I think Poitiers and Agincourt passed into English lore not much because of their decisiveness but rather, so much like Thermopile, because of what the battle told their respective societies about the power (real and imagined) of their values. The longbow became the heart of the story not because of its effectiveness but instead, as the sword is the symbolic weapon of the nobel, it symbolized the power the common English man.
Well, first of all a real longbow can penetrate steel armor, as long as the arrowhead is made of forged, hardened steel. Real longbows were made of yew and were really big, like taller than a man. They were very expensive and took special training of many years to use properly. In the right hands they were incredibly powerful and the arrows could penetrate armor. Also, crossbow bolts can penetrate armor.
As you say, arrows were mostly ineffective against knights in the general case. This was because many archers did not use real longbows, and because if the arrow hit at an angle, it would just glance off. You would have to get a good hit, and the problem is to get a really good hit you would have to be close to the knight--not a healthy place to be for an archer.
However, a good company of real, trained longbowman was absolutely deadly, even against knights on heavy cavalry, and this was clear at the Battle of Agincourt (1415). The Chronicle of Enguerrand de Monstrelet says the following:
From this it should be clear that good archers were effective against knights.