Actually that's a over simplification, By even the early Medieval European warfare featured the bow, the crossbow, Pike men, Trench and siege warfare even artillery.
Remember Much of Roman Warfare was left across Europe and picked up by any number of Chieftains and warlords. Even the armies of Europe were not huge lines of Knights. Knights were the Officers and cavalry. By the later Medieval Firearms and cannon. The Sword was In Europe a Sign of prestige. Any blacksmith can make a battle Axe, a Spear, A war hammer, a mace. Many of the weapons used in the hundred years war were originally farming implements
TE, I would degress, European medieval warfare is mainly built on pitch battles and sieges. I have not suggested that Europe had huge lines of knights nor must knights use swords. Lets use men at arms and knights to denote armored men with close quarter weapons; most European armies except the English were made up of men at arms. The French, Austrian, HRE, Polish armies were predominantly men at arms and knights on horse back.
If we look at contemporary Asian armies: Mongol, Huns and Manchu are all horse back archers; The Sung/Ming/Korean Army was as much as 30% hand cannons, repeating cross bow with poison darts, hand grenades, rockets, fire lances, etc. and you have the typical swords, bows, triple bows, catapults etc. Vietnamese, Burmese, Thai armies were mainly skirmishers. i.e. I would make the case that Asia is much more diverse than Europe in terms of warfare
Also, it depends on when you draw the line for medieval, Rome to me is a bit early, so around 500 AD after the fall of western rome and when eastern rome became Byzantine would be a good starting point for the dark ages, and the birth of the anglo-saxons, Spanish, France, Holy Roman Empire at around 1000 AD would be more suitable to be medieval period. Likewise, By Elizabeth I, we are well into the Renaissance in the 1600s. There are overlaps in the time frame, but when Harold fought the Vikings in 1065 and lost to William in 1066 was really the transformation into the medieval, and till the end of the hundred year war in 1453, Dates are flexible, but warfare within is pretty much consistant in terms of pitched battles with men of arms.
Yes and no. The Swords were not blunt. they had edge and you can handle a sharpened edge with your hand with out cutting off your finger as along as you don't slide it. Also there is some evidence of varying degree's of sharpness on some European Sword edges The first say half foot of the blade being duller then the last. Many historical manuals state the blade should be be sharpened Some stating very high levels of Sharpness other swords take designs that add features like a Ricasso which was more or less a portion of the blade intended to be gripped.
Swords in Europe needed there edge They were not just for Knight on Knight fighting and they were not just for war. Dueling rarely involved armor, Also Knights in there Earliest form were more or less Glorified Mob enforcers used to gather protection money ( Taxes) for the noble Lord.
Well sharpness is relative, simply, broad swords are not as sharp as Jians and Katanas. It is a simple difference of preference, Both asia and europe had crucible steel, Damascus steel, Wootz steel, folding, forging, twist core etc.
European swords are mostly twist core or forged, while asian swords are mainly folded and forged. Twist core provides better shock absorption, but cannot retain as sharp an edge - and it was a choice and preference as the technology was available to both continents.
Try hundreds of knights, Knights were The elite class They were in the service of the Royal Court and could retire to become a Noble lord. European Culture was at that time a Caste system with Nobility/Royalty being in the warrior Cast as well as Religious cast. Knights were the elites Trained in house from childhood for battle tested and sharpened in skill in the War games of the Joust, below them were Mercenary bands men who were experts in military war craft who hired out there services to the Lords of Europe to fight there wars. Example the Reisläufer Swiss Mercenaries who were hired on as whole regiments. Then Came the infantry More or less Peasants paid to use there farming implements for war. Some used Axes, and pikes and Even hacking blades. these grunt infantry from time to time would join the Mercenary bands choosing the life of Travel and pay for war as opposed to the Abject poverty of the Feudal farms and civilian life.
As i said, it is a question of scale, Lets quasi equate a samurai as well trained as a knight and as well provisioned. Tokugawa deployed 164,000 samurai and men versus Toyotomi with 120,000 samurai and men; about 100 years earlier, Ming deployed 500,000 men against Mogolia in the Tumu crisis; the largest internal european battles were around 30K men per side, and even the coalition against the Turks in the seige of Vienna is around 100K europeans vs 90-300K turks. Most European wars are fought with armies in the thousands. An example is, the Landsknechts mercenaries are around 12K men in total, the swiss pike is a total of 20K men, the Genoese crossbowmen is less than 10K men total. the Khitan mercenaries around 40K men, the Huns were around 100K men, the wokou pirates/merc numbered 400 ships, and more than 200K men. Scales are simply different.