"Longbow proved to be far more useful than crossbow on the battlefield" how? For one archer you could train thousands of crossbowmen, that's why crossbow pushed bows from battlefields because of massive difference in learning curve
Crossbows were considered to be "unfair" in chivalry so its use was less glorified in Europe, but it is widely used by mercenaries. Ain't no way that longbow was more useful than crossbow. English longbow was another beast and the only big win was in Agincourt.