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.