Blunt blades can cut almost as well as a sharp sword. Ever seen the effect of a katana vs a blunt bastard sword on a rolled bamboo mat - same result. And for medieval warfare, cutting was pretty damn useless since chainmail would basically negate any cutting effect a sword might have. Stabbing ftw. Slashes would more often result in bruises and broken bones, which of course also hurt like hell.
//Incidentally why I find movies of medieval times where everyone slash each other to death while wearing chainmail to be so funny. The technique used at those times were not slashing moves when aimed at armored bits since that would not result in killing blows, stabs on the other hand. And slashes at exposed flesh.
I find this topic to be getting more interesting - more history!!
There is a difference between a blunted blade and a club carved from a boat-ore in the shape of a katana; do you think that said wooden sword would cut through the same bamboo mat? I would bet on the answer being 'no'.
But another sample of technique requiring a specific weapon would be a draw-cut. using your example, a sharp sword can be used to effect a draw-cut, where a blunt blade would require percussive force.
As for medieval movies having slashes being portrayed as the typical death-blow attack, well, it's actually more of a hacking-style motion and that's because most of those swords didn't have very sharp points either. That and those are movies; where a movie portrays two hours, it shows events typically of several days, weeks, months, or years.
And like a real fight, it would more likely be several chops depending on how heavily armoured your target was; a layer of chainmail over quilt for instance can leave limbs broken and links weakened, meaning that the second strike would chop through the remaining protection and dig into flesh and bone.
Meanwhile, is a more Roman-era style of movie - those swords were really sharp and their armour was only leather. That said, the metal those swords were made of would quickly dull against hard-surfaced armour like splint or plate, and because of their size, they just don't have the weight to bludgeon their way though. And layers of chainmail would prevent the hope of attacking weak points with stabs because the blade itself is too broad.
This brings about more specialized blades, like a rondel dagger, able to punch in through joints and the like; they're great if your opponent is in heavy armour because the dagger is smaller and faster to use, and its greatest benefit is for punching at the weak spots. But against a warrior in leather with something like a Gladius, and you see the advantages removed; the dagger is not quicker than the short blade of the Gladius, it isn't longer, and the "advantage" of punching through weak points in heavy armour doesn't exist because the opponent is not wearing heavy armour.
This brings about full-circle my point of this being a game; in real life, all of these weapons had their advantages and disadvantages, their uses and their eras; meanwhile in this game we've got weapons pointedly described as being similar to a real-world item being relevant against things that they should not be relevant against. For instance, most of the effectiveness of various power-weapons comes from the power-field, but ignoring that and focusing on primitive weapons modified with the Mono ability (or similar for blunt weapons), you'll note that these weapons retain a certain effectiveness despite being used for wildly different things in real-life.
An instance of this would be a "mono-sword" and a "mono-axe"; the axe is statistically better at punching through armour because of how the armour mechanics work in this game; it's not because the axe has a better penetration value, but because it does more damage. See, in this game, the description, no matter how close it is to a real-world item, can be clearly and demonstrably shown within the game as being a work of fiction. Sure, some things will hold a high degree of verisimilitude, like with swords typically being balanced and axes and clubs typically being unbalanced, you've also got your works of fantasy, like the above situation with the mono-weapons.