I am going to open with the fact that you will need someone else to answer questions about medieval arms and production.

You touched a little bit on the answer in your post, namely that Roman’s had a higher capacity to produce arms and armor than medieval kingdoms, and this is true for many other states in Antiquity. This stems from the higher economic capacity of both the economy and the state. Put simply, the Roman state & economy was probably more complex than anything in the west until around 800 years after it’s decline and demise in from the 4th – 6th Century. Roman society was highly complex and economically diverse. It supported cities large urbanized populations that are not matched in Europe until the industrial revolution. Supporting the army and urban populations made up pretty much the entire Roman budget. To support cities, you need to be able to bring a significant amount of goods in to the city to support the population. That means both food and consumer goods. The most famous example of this is the Annonae trade and the grain dole in the city of Rome itself. A massive project that started in the 2nd Century BCE, the state sponsored cheap and free grain to certain citizens all the way until at least the 5th Century. That doesn’t even cover the massive commercial economy that is possible when you have large urban populations and specialized production.

You may have heard of Monte Testaccio in Rome, which is a man-made hill and garbage dump of pottery imported from what is now Southern Spain in the 2nd and 3rd century. Mass produced pottery, which was also used by Roman peasants, has been found all across areas of the former empire. We have found the produce from a mass production facility near Millau, France in Spain, Tunis, Morocco, Algeria, Italy, England, and Germany. The Roman State, and private traders, were shipping millions of goods across the empire towards commercial hubs. This isn’t specifically military like you asked, but it goes to show the level of commerce in the Roman empire. People had the ability to mass produce goods, ship them across the empire, and keep in business.

The army itself was supplied by state run manufactories, fabricae, which could provide weapons and clothing to all the men on the frontiers. This included building said goods, and then shipping them to the needed location. Generally, these facilities were close to the frontiers but the ability to mass produce weapons and get them to the correct camp shows a significant state capacity for organization. There were state sponsored production facilities all across the empire. Even in the 5th Century, a soldier on the frontiers could have been supplied by several different factories.

That complexity, interconnectedness, and capacity disappears from the 5th to the 7th Century in Western Europe. There is debate over how much the overall population actually drops, but Mediterranean trade, especially in the west takes a nosedive in the 5th & 6th Centuries. The urban population drops significantly, several million urban residents in the west disappear and there was probably a drop in the rural population as well. There are recording issues with population, estimates but the overall population drop at 25-33%. We even see significantly lowered coin production as well. An archaeological dig at Yeavering found pots in the tomb of a Saxon King that was of lower quality than the mass-produced high quality, if basic, pottery that was available for many in the empire. Britain is an extreme example, and the regression not as total in other areas of the post-roman west. High quality good remained available for the extremely wealthy, but mass production basically disappears with the economic collapse of the 5th - 7th Century.

I’ve focused a lot on the economics side of your question, because it really is the only way to show the extreme drop in the level of complexity in the Western Mediterranean. Some aspects survive for multiple centuries, but overall economic activity falls off, population drops, and states get significantly less complex. Why were higher quality weapons limited to the wealthy? Because the entire economic and social system that paid for, armed, fed, and clothed the Roman Army slowly disappeared in the West and the states that rose up in the aftermath never had the capacity to do anything close what the Roman state could.

In terms of mass production or production I think 'nickelfldn' answered it thoroughly. In terms of medieval warfare I could tell you that they needn't mass produce swords.The use, or not, of a weapon is heavily tied to the way you conduct warfare. The sword is an expensive weapon to make; it requires much more steel that a spear, or mace and of course requires a specialised craftsman, swordsmith. It has the advantage of being a versatile weapon able to be used in the hills of Spain, forests of Germany, mountains of Illyria, deserts etc. However it has several disadvantages. It has not the reach of the spear, but most importantly it can't pierce through armor, unlike the movies. Now medieval armies were rather small compared to roman or Hellenistic period armies. Their composition although certainly varied depending on the region could be described as such: The mounted elite warriors, lords and knights heavily armored, a small core of professional men at arms maintained by the said lords and knights also armored, and the rest and majority were peasant levies raised by the lords' demense who if they were lucky were given a helmet, shield, spear and a cross necklace for good luck. Their effectiveness in combat was doubtful at best. In an ironic way we can see that even in the medieval period all soldiers, at least professional soldiers, had access to swords.
Still as medieval armies became more permanent, the states became more centralised, the sword was never to regain its position as primary weapon which had in the roman era. And that is because it was ineffective against armor even the cheapest armor the gambeson which after the Hundred Years War would probably have be worn by every combatant. The use of the sword in roman era is tied to the way the romans conducted battle as opposed to the phalanx the greeks and most other people used in some extend. Perhaps it came from the necessity of fighting against different people in different environments. They had to be versatile and they also had the advantage of having a state capable of supporting their army. The roman army was the first real professional army. Nor the spartans not even the Macedonians of Philippus and Alexander were a professional army in that regard. People tend to think of the roman army as invincible but that was far from the truth. In many cases the legion faltered even in battles that were finally won, Cynoscephalae for example, or Magnesia. The romans however had the luxury of being able to lose one, two, three or more battles, luxury Hannibal, Antiochus and several others didn't have. What made roman expansion possible was diplomacy, trade and logistics not the armies. The armies were just there to confirm the conquest. Well the last one is my personal opinion you can make what you think of it.
In the middle ages the mode of fighting for infantry reverted to the phalanx-shield wall where the spear would again become the primary infantry weapon. Read about battles in the early Middle Ages: Tours, Haestings and read about battles of the romans. You will notice many differences in tactics, logistics,army composition, numbers etc...
So to close and answer your question in terms of warfare is that medieval states Neither Could as 'nickelfdn' said nor Needed to mass produce swords.
As an afternote however, it is interesting to see that although mass production of swords was not taking place we know for a fact that the Arsenale in Venice at its height had the ability to manufacture a ship every day so the concept of mass production might have been uncommon but not impossible even in the middle ages, given enough money and necessity and as our friend Euripides says: "Nothing has more strength than dire necessity".

I hoped this helped you or at least gave you a different perspective. I'm sorry for my English I don't write often so I've become rusty.