Terran, I was thinking along similar line as you, perhaps the difference in armor is the answer. But I was think more about mail armor (popular in Europe but not in China), which is resistant to cuts but vulnerable to thrusts from sharp pointed sword. But on the other hand, lamellar armor were also very cut resistant, so why the move to Chinese single edge sword? Then you have Arab/Muslim armies that used mail armor very extensively but also used single edge type sword.
Full plate armor is a late medieval development in Europe, so that doesn't really explain about a thousand years history of preferred type of sword in Europe.
If you look at this chart of sword pattern development in China, you can see that early Chinese single edge sword (dao) have what looks like very strong tips that looks like they would be good at piercing lamellar type armor.
Well, if I remembered my history lessons correctly, the Dao became the predominant side-arm weapon during Han mainly due to cavalry. The main weapons of the infantry where pole arms and the cross-bow, while the main weapon of the cavalry where Dao.
The main side arm of the warring states were the double edged Jian because stabbing was a more efficient way of killing than slashing, It was the time when mass infantry was the dominant force between all sides, and most battles involved some hand-to-hand combat on foot. The Warring State jian's were predominantly used for stabbing. The thickness of those bronze and iron jian's actually varied in a wave pattern designed to absorb the shock from the blade stabbing into an hard object, like through a body, or armour, so that the blade won't snap.
The main enemy of the Han empire where the Xiongnu (Huns) on horse back. Cavalry became the main strike force, and the infantry mainly used to provide cross-bow fire and hold positions. The primary method of killing on a horse back is slashing. The single edged Dao design allowed a heavier and more robust blade, that is required to withstand the extra force inflicted from both the rider and the momentum of the horse. Further more, a Jian is more expensive to make.
One has to note that most of the Chinese popular culture and Kung Fu related stuff has nothing to do with the actual history, and most are just fantasy:
For example, Guan Yu NEVER used a Guan Dao (crescent moon Dao)... the earliest official record of the existence of a weapon in a similar form of the crescent moon pole arm was not after several hundreds years later.
The famous "ox-tail" Dao (i.e. what is commonly known as a Chinese broad sword) that is commonly seen in Kung Fu movies and Kung Fu competitions is in fact an almost exclusive civilian weapon that only came to existence relatively recently. None of the Chinese militaries of all dynasties ever issued such as their weapons. The standard issue swords of the Ming and Qing armies were "swallow feather" and later "willow leaf" Dao's, which has narrow blades curved in a similar fashion as the katana, compromising between the effectiveness of stabbing and slashing.
A lot of the so-called famous Chinese weapons were nothing more than modified farming tools used by civilians to bypass government weapon regulations.