Here's more, with the key point based on witness accounts.
But at this moment the archers, taking their hatchets, swords or other weapons, penetrated the gaps in the now disordered French, who could not move to cope with their unarmoured assailants, and were slaughtered or taken prisoners to a man.
This neglects to mention that "stakes and spears" were prominently mentioned in what the archers used when they waded into combat. It also asks us to believe that unarmored assailants, given similar weapons, can slaughter armored ones. If that's true, why did anyone ever bother wearing armor?
If the whole French line could only kill 113 English men-at-arms in the whole battle, while fighting in heavy armor, it's a stretch that lightly armored archers, using hatchets and swords, could kill 5,000 equally armored French men-at-arms in a similar timespan, much less with a 50:1 or 100:1 kill ratio. I'd suggest that in any poorly organized attack over difficult terrain, a lightly armored man who doesn't have any reach advantage over his better armed adversaries would normally result in a kill ratio that favors the heavily armored knights. When I see a lopsided kill ratio like that, I start thinking that one side must've been able to fight while staying out of lethal weapon range of the other. That makes me think that a pole arm formation executing an envelopment from the flanks requires the least stretch of imagination, as even with long wooden anti-cavalry stakes the archers could've killed or knocked down and pinned the French men-at-arms, finishing them off with hatchets if need be. If would also explain why the French couldn't just back up, as they were being enveloped. That action would certainly pack the French line together, compressing it into a useless mob no matter how good the French planning had been.
If you look at this map of the second stage of the battle, you can see it's perfectly set up for such a maneuver.
Here are some links on schiltrons. Stirling Bridge
Falkirk
Another Falkirk link
One reading of the Stirling Bridge battle is how Wallace goaded the English into a precipitous charge across the bridge, where the schiltrons were to cut them down. The technique was further refined by Robert the Bruce, who famously used it at Bannockburn in 1314.
About half-way down in the last paragraph of this we see Robert boldly riding with his crown on, making himself an obvious target and frankly some sweet bait, drawing the English into another precipitious attack. Then Robert unleashed his schiltrons.
[color="blue"]Edward II thought that the Scots were terrified of a head to head confrontation, and when they began advancing, he was convinced that victory was in his hand. He ordered the Earl of Gloucester to launch a massive full-frontal attack, but the inexperienced Edward did not realise that he had fallen straight into Bruce's trap. Edwards forces were positioned on the boggy carse, as Bruce had intended, and were also squeezed between two streams - the Bannock Burn and the Pelstream Burn. On launching the attack, the Earl of Gloucester was met by the advancing schiltrons of Bruce's army. The circular hedgehog-like formations of Scottish spears cut down the English cavalry and repulsed their advance. The Earl was killed, and Edward had lost one of his few battle tacticians [/color]
The technique is to goad the enemy into attacking at a position (Stirling Bridge) or on terrain (boggy ground) that hinders their movement, where they can be pinned and cut down by a phalanx of overlapping spears, which are best unleashed as a complete surprise. If the schiltrons attack from the flanks they'd crush any opposing formation.
The English and Scots kept up with the spear combat, here recorded in The Ballad of Chevy Chase, about a 1388 battle, just 27 years prior to Agincourt.
[color="blue"] Of fifteen hondrith archars of Ynglonde
went away but seuenti and thre;
Of twenti hondrith spear-men of Skotlonde,
but even five and fifti.
[/color]
So it makes me wonder whether the English archers didn't "just wade in", that their maneuver was part of the English plan, which depended on goading the French into a frontal assault on the English line. They also eliminated the French cavalry first, which would present the only threat to a wheeling flank of English archers. It just smells like something carefully planned, especially given the results. However, given the mud and confusion, to the French it may have looked like a confusing action executed on the spur of the moment.
Unfortunately I haven't had time to dig into the first-hand sources in enough detail to flesh out this idea.

