The English "long"bow is by far the most overrated medieval weapon. Comparing it to a machine gun is not very wise. Even thousands of archers were not able to stop an enemy and repulse an attack what is what a machine gun can do. As one example you can take the battle of Towton when the Lancastrians had wasted their arrows and were free targets for the Yorkist archers. The arrow showers forced the Lancastrian army to attack but were by no means devastatious to them.

"Long"bows were simple powerful self-bows, nothing special compared with asian composite bows but used throughout Europe since the stone ages. Special were in case of the English only the men who draw the bows because they were very experienced and well trained and could therefore draw very strong bows up to 160 lbs. But even then the bow was no wonder-weapon. The French f.e. won a lot of battles against English armies with strong numbers of bowmen during hundred years war. It depended on situation.

Bows proved rather bad against plate armour. They helped indeed much to win battles against bad equipped foes or in certain special situations (Crecy, Agincourt). But at least from 1450 on the best time of the bow was gone.

Puuh. That said I concur with the assertion that longbowmen are too weak in some aspects in M2TW compared to crossbowmen. The powerful crossbows capable to punch through some forms of armour (crossbows were poor performers against plate, too) had steel bows with pull weights of 500 to 1000 lbs. and were not easily drawn. If you give a longbowman 10 arrows per minute (which would be rather fast and exhausting shooting) a crossbowman should be capable to fire max. 3 bolts per minute. The longbow should have more reach than crossbows, similar to that of muskets. The performance of bolts and arrows in M2TW is generally too high but with the abundance of plate armour in late game it would be unfair to lower it. So mod the forementioned two things and you can be satisfied.

By the way: the Aztecs are unrealistically strong compared with old world units. But that is for fun, otherwise it would be boring.