I include crossbow bolts under 'arrows' seeing as they are fundamentally the same thing, there are also historical accounts of bodkin point arrows penetrating plate armor when fired at close range, and modern recreations and computer analysis has
backed that up.
Pole-arms also includes the Halberd, a weapon
specifically designed for unseating heavily armored knights and then wrecking their shit, while still being effective against spear and pike formations.
The primary advantage of the crossbow over the longbow is that training a man to use a crossbow effectively in battle is a task of a few months to half a year, much the same as with a gun, whereas training a man to use a longbow effectively in battle is a task of a literal lifetime that must begin when the man is a child and takes decades of work and expense. Crossbows are thus viable peasant militia weapons and can be deployed en masse, whereas longbows are not.
While there are other differences between the two forms of 'bow' weapon, the difference in skill means that orders of magnitude more crossbows can be used effectively compared to longbows and the sheer difference in numbers is by far the most overwhelming advantage; something the Chinese knew
very well thanks to their ridiculously early invention of the repeating crossbow in the 4th century BC: The repeating crossbow might not have had the same power behind it as the much later European arbalest, but combined with a bit of poison on the bolts, the sheer
number of projectiles a bunch of peasants on walls armed with repeating crossbows could throw out made them fucking
impossible to assault.
Real-world battle is a complicated and chaotic affair, 'primary' sources of certain kinds of damage might be the
best, so to speak, but 'secondary' sources (ex; an armor piercing dagger to the armpit) can fuck your shit up just as fatally, and
were used. And ultimately, at the end of the day, if enough people with swords beat on an armored knight for long enough, they'll probably kill him
eventually even if it isn't even remotely the most efficient way to do so.
There's also the reality that not everyone had the best of everything in the best condition all the time; wear and tear happened, low quality materials and equipment happened, shit happened, all of which can alter the outcome.