You separated all the pros I listed for the axe instead of thinking about how you get all that in one package. It's not about this one sword is faster, and this other sword is cheap, and then there is this sword that does more damage. It's about how an axe does more then those sword but with it's only con being less reach. Also the cheap upkeep part was about having an axe that deals more damage than most swords at almost a fourth of the cost depending on which axes and swords you are comparing.
Military Cleaver, one of the highest cut damage swords slower than all other one handed axes, doesn't break shields, deals the same amount of damage as the axes, and it's heavier. Only pro is that it's longer.
But if you want a fast 1h weapon you are going to have to sacrifice some damage, but the axe has speed and damage, 98 speed can be spammed easily. I'm not sure if you've used many 1h axes?
If you want a long 1h you are going to have to sacrifice speed, usually, and they still won't deal as much damage. So there are trade offs.
The Iron War Axe deals the most damage out of all 1h. So obviously it needs a trade off speed and reach. Grosse messer isn't good for fighting cav, or s-keys.
In the end it all comes down to which weapons you are comparing because, thankfully, we have a nice variety of one handers that have pros and cons over eachother. You want more damage, sacrifice speed or length, you want more speed, sacrifice length or damage, you want to be fast and high damaging and have the ability to break shields, sacrifice length.
Axes have their own niche to fill and they do it well. Obviously they will be worse in some situations and better in others, just like all weapons.