They're supposed to be here to clean up lightly armored infantry (at the dagger level, it's actually "unarmored" infantry), but they don't perform so well in that job because even with the increase in speed, their total damage is lower.
Part of the problem with only using the highest level weapon is that everyone gets attacks based upon their movement, and not based upon the speed of the weapon. There are no real downsides for using the inherently more "slow" weaponry. Similarly, the "fast" weapons make you move faster? I don't walk more briskly when I'm carrying a knife than when I am carrying nothing.
If attacks used a % of your movement points rather than an integer value, you would see a much greater advantage to "fast weapons" over harder hitting ones. Fast weapons give you extra opportunities to deal a lesser level of damage. At this point in time, you already get pretty much 3 attacks on every unit out there, 4 if you have the right faction trait, more if you are riding a horse, etc. Similarly, it would be much easier to balance the really heavy hitting weapons. Make them slow, and use 75% of unit's total movement.
So, 4 Attacks at Str 9 (32 max damage potential) , or 5 Attacks at Str 5 (25 max damage potential)? Not really even a choice.
If you reduce that to 50% movement for a single attack or 33% movement for a single attack, the comparison becomes more attractive.
2 Attacks at Str 9 (18 max damage potential), or 3 attacks at Str 5 (15 max damage potential). This has more considerations. Are you fighting lightly armored things? That extra attack may be more worth your while.
Let's look at the super heavy weapons in this one as well;
Great Scimitar is 45 Attack, right? Ok, well that's a "slow" attack, so 1 chance to hit and do 45 damage. As compared to a broadsword where you get 2 attacks at 18 (36 max damage potential). That makes the big big weapons great for blowing a single unit's head out it's tookus every turn regardless of armor, where the weapons with less damage per strike give you the ability to attack several units a turn.