I'm not sure about specific reasons, but the kopis and falcata are not that effective in my opinion. The Romans themselves knew it, and they employed short thrusting and slashing gladius, which is basically superior to any other sword in the world when correctly utilized. The slashing technique of sword attack has quite a bit of disadvantages, such as slashing was and is never as effective as a clean, strong thrust. Only the largest swords were effective in this mode of attack. Few types of armour can withstand a thrust on the other hand. The slashing motion is also difficult in the crowded heat of battle, and it exposes your body to a counter-stroke as you move the shield aside to slash. The Romans knew this, and so they used the gladius, instructing the legionaries to use their large, rectangular scutum to cover themselves as much as possible and attack in brief clean thrusts to the side of their shield. All this time their body was unexposed.

However, I doubt the idiots during the Dark, Feudal and High Medieval periods knew all that...