Nitpick: historical longbows could pull around 180-190 lbs. some even 220, well in excess of the measly 100 you describe. The bodkin arrows are all well and good but it is absolute nonsense that arrows pierced platemail on a regular basis. Even regular mail (aka. chaimail if you are a D&D type of guy) was nigh-on-impervious to arrow fire unless done at point-blank.

Crossbows on the other hand had a much heavier pull, starting at 220-280 lbs. for the "light" military variant and going up to 900 lbs. (!) for a sort of portable ballista requiring two men to operate, as well as a crank mechanism.

The success England had vs the French in those aforementioned battles was not because they mowed them down with ease (like Age of Empires II or Hollywood movies would have you believe) but because the French were terribly disorganized and poorly lead. Still, when the battle reached melee, the Longbowmen threw their bows away and joined the fray with their mallets or other sidearms they carried. The majority of the battles were still won with hand to hand combat! Other factors such as terrain and weather played a huge role as well, but that's besides the point.

Now, developing the back and shoulder muscles needed to reliably use a 180-200 lbs. longbow basically requires one to be using it from the earliest age possible (11 or so), whilst any able man grabbing a crossbow could learn how to use it in a few months time - hence the Pope banning it (too hard to control, diminishes the value of mounted men-at-arms who were more or less controllable by the nobility and church)