I get a little carried away on this topic, it`s pretty close to my heart. Again, I want to make clear I intend no disrespect to other martial cultures.
As far as curved edges enhancing slashing effect, you`re dead on. You can see on a lot of early bowies how the entire edged is curved, though to a lesser degree than the belly.
First off, a little historical clarification. If you`re talking Turks, Mongols and crufiorm swords, you`re talking Attila and The Crusades. The difficulty Western armies found facing the both the Mongols and the Saracens were the fact they had mounted bowmen. In fact, the Mongols were so connected with their horses they were rummored to be incapable of walking. Moreover, during the age of the cruciform sword the primary weapon of the mounted warrior in Europe was the lance, which will out do a scimitar any day. After the age of the lance came the age of the saber, which was also intended to be used for making point in a cavalry charge. Check your history, the Classical/Dark Age/Medieval warrior had no trouble overcoming his enemy in a toe-to-toe fight. I will readily concede the mounted bowman was an excelent idea. After we invented firearms(late 13-early1400`s, shortly after The Crusades), we came up with the dragoon. But let`s avoid the "firecracker" debate. Fireworks and firearms are functionaly different. I`ll have to check, but I`m pretty sure it ends up the Moors and Saracens weren`t using radicaly curved blades during The Crusdaes anyway. Also, non-European, metal-sword-using culture that independently developed cruciform weapons; the Tuareg. I don`t think anyone would have the audacity to claim the Tuareg were not fierce and proficient warriors. If you don`t know them, they`re African. Their name comes up with Timbuktu a lot. For that matter, check out the Indian pata, it`s got a wierd hilt, but the same blade-style. Hey, actualy, there are many cruciform swords in Near-and-Far-Eastern cultures, or at least ones that have the same style of blade we`re discussing, hilted according to local fashion.
Now, there were two well-known curved swords in Europe, previous to contact with Far-Eastern cultures. The marchione(sp?) of the Greeks, also known as the espada falcata to the Spanish, who originated the style(they also brought the Romans the gladius, by the way). Note that this sword-length weapon is the source of inspiration for the kukri, universaly regarded as the de facto chopping blade. There was also the falchion. A wide-bladed, curved weapon popular for litteraly centuries.
Two things you`re not taking into consideration, "cruciform" refers to a double-edged sword with a crossgaurd that only extends fore-and-aft of the hand(knuckles and wrists). The numerous variations on basket hilts are not known as cruciform. They supplanted cruciform as they provided better protection. Second, the fact that many true cruciform swords are tapered off towards the point. This also makes the edge come into contact with, well, whatever`s in it`s way, at an increased angle, increasing the effectiveness of the slash while not deminishing the ability to thrust, but rather increasing it. You must also keep in mind that these swords are nowadays called broadswords because, they are. Their increased breadth allows them to have two edges and still maintain a thin cross-section for cutting. Like a machete(descended from the espada ancha, I believe), where the slim cross-section allows it to slice with ease. Obviously, this is MUCH more pronounced on a machete.
I think upon reflection you would choose to withdraw the idea of the evolution of arms and armor in Europe as slow. I`m not going to go too much deeper on this one, it`s readily apparent.
I have no problem using a double-edged knife, or axe, for utility. Their are certain things you can`t do, split wood or use it as a draw knife, but hell, even Daniel Boon used a double edged knife for utility. Now, you wouldn`t saw he didn`t know what he was doing, would you? I have been playing around with several(including one that looks a lot like and espada ancha/the sword you came up with) fascimiles of swords made from plywood to get a feel for what it`d be like to lug around. Honestly,I would love to cary a sword, but considering an M-16a2 rifle is the primary weapon of the American soldier these days, and the sword would not see a whole lot of duty, I find it to be a bit out of place for me. I`ll keep the knife though!
Closing statement;
I think that the reasons why the Oriental styles have supplanted the Western styles are threefold, 1. the "familiarity breeds contempt" issue, 2. they are more readily accessable(i.e. the ever-present strip-mall karateka), 3. the Western arts have moved on. As much as I hate to say it, blades and bare-knuckles are not the most-effective weapons today. Martial arts are "martial" as they are, or were suitable for war. The reality(though it blows my Don Quixote romanticism out of the water), is that the firearm is the martial weapon of the Western, and to be sure, the Eastern, well hell! The whole world! Upon reflection, I believe American Gun-Fu, the pistol, rifle, and shotgun-craft tought at places like Gunsite and Thunder Ranch represent modern Western martial arts. However, this does not diminish my desire to re-vitalize, and train in, more "traditonal" Western styles. In fact, I am presently engaged in translating a fight manual that deals particularly with the blade-arts of my ancestors.
Like I said, this issue tends to get me riled-up, so if I wrote anything that is derogatory or insulting beyond simple criticism or point/counter-point that is necessary in any debate, I apologize. I bear no ill-will towards Eastern-stylists, non-Westerners, or the party this reply is aimed at. Peace.