You are wrong about the armor issue, though. With the introduction of heavy plate armor, crossbows became obsolete against heavy Western cavalry. Plate armor was *tested* and “proofed” using crossbows with AP quarrels at point blank range. Yes, eventually armor progressed technically by the end of the crusades and after that, but the Pope had still stated what he earlier stated during the start of the crusades. Islam did not use the crossbow much. The Western world had Knights while the eastern world had Cataphracts. During the early to mid crusades the Cataphract armor was superior to that of the Knights Templar and the Knights Hospitalier. Just look at any drawing of those Knights and compare it to any drawing of the Cataphracts from that time period. The Eastern world had more advanced armor during that time because Islam had conquered and assimilated a more advanced Eastern Rome as compared to the Germanic tribes which had conquered a less advanced Western Rome. While the Germanic tribes assimilated Rome earlier, it took hundreds of years to advance out of the dark ages by converting to Christianity. The Middle East also adopted the military advances of the far east at that time, before Western Europe (gunpowder for example). Eventually however the Mongols arrived and made the heavily armored knights and cataphracts relatively worthless. Even with their most advanced armor.
Yeah - armor doesn't help much when you're up against a people that grow up on horseback and are horse-archers of such skill that they can target and hit a helm slit. Add to that the fact that they were the first army to use what we'd recognize today as "modern" tactics and strategy and you can see why they went through almost all the Islamic and European armies that stood before them like a hot sword through butter.
Imagine if you will, a group of Islamic or European cavalry of the 13th or early 14th century. A band of small men on ponies gallops into range, looses some arrows at this force and then gallops away with the other force in hot pursuit.
Now, keep in mind that cavalry of the era tended to attack "on line", i.e., side by side abreast, in a somewhat disorganized charge. After the initial charge, it devolves into a swirling melee of men and horses - but pretty much it's all or nothing at once with such tactics. Roman cavalry tactics weren't that much better.
So, the Christians or Islamics crest a hill, and this is what they see on the other side:
The Islamics or Christians would suddenly discover that they were pretty much a bunch of amateurs up against the first true modern army in the world. Easy to see why my ancestors pretty much steamrollered most who stood against them.
The irony of it was that the Khan would have been perfectly happy to let the Islamics and the West alone, but NOOO, the Islamics had to be their usual chop-happy selves; they executed his friend, the ambassador that he'd sent in peace to establish trade.