Yes, I'd certainly expect the longbow-fired bodkin to still penetrate the joints of the armor, and certainly to stop a mounted charge, which the episode asserted it did. The show's analysis of arrow vs. armor was far from exhaustive, but did make me question some of my casual assumptions. It didn't claim the longbow had no effect in the battle, just that it had less than is traditionally allotted to it, and other aspects (such as the suction of mud on metal plate vs on cloth and the type of soil on the field, crowding, etc.) were the critical ones.

I liked the way they worked the armor-piercing quality of arrows in the first MTW. Regular arrows had the potential to kill even the best-armed knights, though their chances were much lower, which could correspond to the lower chance of hitting a joint or other weak spot. Against the less-well-armored warriors of the early period they were more effective, though still not nearly as much as the better longbows or especially the crossbow-armed troops. I think this accurately bears out both the protection afforded by armor and the potential of any arrows to still succeed, but moreso for those that are armor-piercing. Also, the horse makes a target more vulnerable, though the armored horses of later ages (or some of the eastern civilizations) have at least some protection.

I might perhaps adapt it so that regular archers have no armor-piercing bonus (though still the potential to kill armored troops), longbows are armor-piercing against chainmail (which would surely be the majority of armor, and crossbows/arbs against all armor.

Ajax