Pure edged weapons certainly is a viable and useful strategy, but it still shouldn't be strictly superior to pure blunt weapons. So I'm surprised your first party showed such one-sided results. Let's look at the numbers.

Maths
If a weapon penetrates, full damage is used for dice rolls. If it matches penetration, it does 1/3 damage. If it doesn't penetrate, it does 1/8 damage.

Stats
Longsword does 12 damage at 3 penetration.
Shortsword does 8 damage at 4 penetration.
Military hammer does 8 damage at 5 penetration.

5 penetration matches plate.
4 penetration matches brigandine/chainmail
3 penetration matches scale.

Early game
Early on most enemies are wearing leather so the longsword certainly is the most useful, no doubt about that. When we're fighting thugs in the dark alleys the longsword does 50% more damage than other options. The downside is that it trains edged weapons instead of blunt weapons, so if we're going blunt weapons we'll just have to deal with the lower damage at the very start, which isn't a big deal. The benefit of blunt weapons is how fast it lets us progress in the next stage of the game.

Raubritters
Our first raubritter will probably be wearing chain and scale with a sergeant wearing scale and leather. There's a 50/50 chance of hitting vitals vs limb.

Assuming our characters have a strength of 35+, that's a +2 to damage. Assuming we've got weapons from Nurnberg or another city with weapons quality of 35+, sergeants and weaker enemies (but not raubritters) have armour quality of 25, so that's another +1 damage for +3 damage total. I'll also only compare the max damage used in dice rolls rather than use the dice roll averages.

Against the sergeant, the weapons will do this much damage:

Longsword = 15 damage 50% of the time + 7 damage 50% of the time = average of 11 damage per hit.
Shortsword = 11 damage all the time.
Hammer = 11 damage all the time.

Against the raubritter, the weapons will do this much damage:

Longsword = 3 damage 50% of the time + 6 damage 50% of the time = average of 4.5 damage per hit.
Shortsword = 4 damage 50% of the time + 10 damage 50% of the time = average of 7 damage per hit.
Hammer = 10 damage all the time.

But the raubritter might have plate and chain, and so the weapons will do this much damage:

Longsword = 3 damage all the time.
Shortsword = 3 damage 50% of the time + 4 damage 50% of the time = an average of 3.5 damage per hit.
Hammer = 4 damage 50% of the time + 10 damage 50% of the time = an average of 7 damage per hit.

These calculations ignored effects from tactics like vulnerable spot attacks, which may or may not be a good tactic to use depending on many complex factors.

So overall, the weapons are all evenly matched against the sergeant, the longsword has the advantage in taking out all of the weaker underlings, and the hammer does more damage against the raubritters. In the raubritter battles, we can certainly make a case for using the longsword to take out the underlings faster, thereby evening the odds, and then switching to the shortsword, but we're still going to be in for a protracted fight against the raubritter. So at this stage we can still make a case for using edged weapons, but we're still in the early stages of the game and already we can see the benefits of blunt weapons.

The problem is that as the game progresses, more of the regular enemies start getting outfitted with brigandine/chain and then the dwarves are all wearing chain. So the blunt weapons leave us in good stead in the tougher portions of the game when the right alchemy and saints aren't going to be available. It takes time to locate the proper saints, recipes and ingredients, especially if we're role-playing and not using spoilers or min-maxing.

Potions
I completely agree with you that Strong Edge makes edged weapons excel in the second half of the game. Even the weakest Strong Edge potion makes the longsword match dragons and demons and lets the shortsword penetrate them. But everything comes at a cost. The cost here learning the recipe, buying the ingredients and making it. Strong Edge is one of the hardest potions in the game to brew, and it requires aqua regia, which is one of the rarest and most valuable ingredients. So for the majority of the game we won't have access to this potion, and once we do, we're looking at an endgame-ready party. That doesn't mean we have to stop exploring the world, but it does mean that we've got many viable options available to us.

The deadly blade potion is certainly also good, but I think you downplay the value of the Great Power potion. The top quality Great Power potion grants +30 quality, which is equivalent to +3 damage. Not quite as good as deadly blade's +5, but nothing to sneeze at either. I think a slight difference is that Deadly Blade's damage adjustments occur before penetration whereas Great Power's adjustments come after, but I could be wrong. It's an inconsequential difference anyway because you're combining Deadly Blade with Strong Edge.

But I rarely use the weapons potions. Why? Because it rarely strikes me as practical. By the time we can brew the weapons potions with any reasonable chance of success, we've been stockpiling enough Essence of Grace potions to heal all of Germany. Essence of Grace is cheaper, easier to make, and more readily available.

And say we get our hands on some aqua regia. What should we use it on? I'd prefer to brew it into Sunburst or Eyeburn potions. Why? Let's look at a few scenarios.

Single battle
Say we lay siege to a raubritter's castle or encounter a difficult enemy.

If we're using longswords, we'd use 4 Strong Edge potions and beat the raubritter and his men quite handily.

If we're using hammers, we could just take the raubritter and his men straight on. Or we could wait until they get close and use a Sunburst or Eyeburn potion so that we can gang up on them one a time to thin out their numbers. Assuming 4 battles, that's 4 potions.

Dungeons
This is where the Strong Edge potion gets better because it lasts for the duration of the excursion. However, we also wouldn't need to use Sunburst that often either since we would only need it when the enemies are in groups. It also has the benefit of reducing our exposure to missiles, and so even if we're using Strong Edge we'll likely use Sunburst to blind the archers while we deal with their meat shields anyway.

Skill transition
Once we get to the stage where we have access to Strong Edge, then a skill transition becomes a viable strategy. We could use guild trainers, but that's boring. Just equip one person with a long sword, pray for them before battle, maybe chuck in an Eye Burn potion, and they'll get their skills up quite quickly. Then move on to the next character and so on.

Conclusion
Pure edged weapons remains a viable strategy with its own nuances, just as the pure blunt weapons strategy has its own nuances. I wouldn't consider either strictly superior to the other. And, of course, a mixed weapons party would have more versatility than the pure weapons strategies. And isn't the ability to role play and employ diverse strategies what gives the game longevity?

Wow, that was certainly comprehensive. The only thing I'd add is that short swords are faster than hammers, so you do get a little extra comparative damage boost against armor that way, especially when unencumbered and berserking to maximize the speed advantage.

I think you're probably right that playing naturally there is going to be a midgame period where blunt weapons really have a chance to shine. For that matter 2 blunt and 2 edged users is probably a good compromise, especially if you direct battles yourself and send people against the foes they're better suited for. I just compressed the midgame to basically nothing, so that meant starting with short swords was more advantageous, and there were minimal intervening fights before I could make all the potions I wanted which sealed the deal for blades. The starting short swords were sufficient to kill raubritters, and once I killed the raubritters and trained I had infinite money and potions with alchemy. Early game: kill 2 raubritters, midgame: use the money to train, endgame: actually start adventuring with the benefit of maxed skills and money on demand.

What I basically did was follow the guides my first game until I got partway through and figured out the system (I quit after disrupting a Sabbat, which was actually pretty hard as I didn't have potions and depended solely on saints). My second attempt, I looked up the details in the cluebook and pretty much powergamed and savescummed to see how fast I could break the game out of curiosity. Turns out that's pretty fast.

With 4 recruits with short swords and a Hanse, it really is realistic to knock over a Raubritter immediately without any preliminary fights or buying any supplies at all. I found the standard method of knocking over thugs at nighttime to be quite boring, so I just skipped that part, and discovered you can easily skip that part, at least if everyone has good starting weapons. All you have to do is eat dinner with the raubritter, then ambush him. If you're lucky he won't even be able to call reinforcements. Make the armored Hanse tank for you by parrying, then the other 4 can dice him up in seconds by berserking with short swords and zero encumbrance. By getting multiple quests in multiple cities to kill the same raubritter, you can easily train up all the trainable skills with at most 2 raubritters, and once alchemy is trained up, your money problems are over. Use a charisma saint to boost your face, buy up reagents, craft 255 of a simple potion in only one day, use charisma saint again to boost your face to sell the potions, repeat as necessary.

Yeah, min-maxing will break any game that actually allows the player to role play, especially if you're willing to reload to get the desired outcomes.

But the beauty of an open-ended game is that once we've beaten it one way, there remain many other ways to play. Have you tried a game, for example, with two atheist fighters supported by two archer characters who can barely move unencumbered in their leather armour?

Otherwise I find that once I get alchemy up and running, all encounters become very easy when you've got Essence of Grace potions at your disposal. Not that the characters would need those potions very often anyway. Hopefully a remake will add more high-tier challenges.