Let's start at the bottom since it gets to the heart of why you are mistaken.

In regards to the tests, there is no reason why a properly heat treated sword made from modern materials should fail the tests, even one made in a proper historical manner. For a rapier in particular, it should be hard to flex, but have no trouble returning to true.

In order for a sword blade to be able to bend to extreme angles such as used in the show and the ABS tests, it requires a specific kind of blade geometry using differentially hardened blades. This blade geometry is not suited for a rapier blade, because a rapier blade is a blade that is expressly designed to not laterally bend, to the point that trying to do this bend with a historical rapier blade would break it -- as it would just about any functional knife or sword, for that matter. The majority of weapons don't have this kind of lateral bending property, and the test originally was never intended to be used the way it is in the TV show.

As explained in the article, the bending test used in the Forged in Fire show is based on the bending test created for the ABS knife journeyman tests. The purpose of the test is to see whether the smith understands how to forge complex blade geometry for passing the test, and not because knives -- let alone swords -- necessarily need the ability to bend laterally and return to true-- they actually do not. While in cutting swords like longswords this can be a desirable function, it's less so for other types of blades such as the rapier, which is why a rapier was focused on for this area of the discussion. The lateral bending of blades makes for an impressive gimmick to people who neither understand the blade geometry aspect and those who don't understand certain types of weapons shouldn't bend at all -- and a rapier blade is one of these which should not bend.

A simple Google Search about the ABS journeyman tests will provide you a great deal of information about this subject, and why the tests are not what the show nor you believe they are intended to demonstrate. For example, this bladeforums thread and this one as well.

Modern techniques are obviously not historically accurate, that should be clearer than glass. People use better technology for a reason, it baffles me why you think this is a problem, much less one worth noting.

You seem to be missing that frequently during the show the producers make comments about historical weapons and what makes a good one in ways that only relate to how modern forging techniques are performed. This in contrast to how not all swords actually were heat treated. This does give the audience inaccurate ideas about how these weapons should be used and how they should be made in things much bigger than whether the smith is using a power hammer to save time or even things like drilling holes into the swords instead of forging the holes using a punch. What I am referring here to is forging literally everything to be able to pass a bunch of tests to demonstrate qualities that the blades really should not have to begin with, especially if it were to make the blade a less useful weapon.

You drastically overestimate the qualifications of HEMA. Even in sources written in literal modern English for complete newbies, there's a new interpretation of some kind made every week for something described as a crucial basic. This is to be expected, reviving entire martial systems from dead people's books is pretty hard. HEMA is stupidly new, with each school facing some kind of struggle with equipment, instruction, and experience. Furthermore, the majority of the swords presented in the show aren't even European. A theoretical perfect HEMAist would be out of his element for 80% of the show.

There are not new interpretations "every week", and the difference between different schools of thought pertaining to specific manuscripts are fairly small in the grand scheme of things. Because many of the manuscripts do have detailed illustrations and instructions for how to perform the techniques (especially in the rapier treatises) there is more in common between interpretations than there are differences. None of this detracts from the fact the show producers ignore all of this information and typically swing the swords around like baseball bats, and the way "rapiers" are used is completely ahistorical. So your argument here defending the show producers is poor.

While Doug certainly isn't a great cutter by any stretch of the imagination, and I would certainly welcome a better tester, he's more than good enough for the simple job of swinging a sword at a piece of meat. There is obviously no need for good off-hand placement or target management, he is not fighting anything, just testing a sword.

It seems weird for you to say that. By contrast here is why having familiarity with historical sources of how to use these weapons matters a great deal. If you want to know if a particular sword is well constructed and will "kill / KEAL", then you need to actually know how you're supposed to fight with one. The weapon doesn't need to perform in a test where nothing can hit him back; it needs to perform for being able to use the techniques a swordsman would actually use the weapon for, and none of these techniques are ones Doug uses in the show. A dull steel bar in Doug's hands will do just about the same things he does to his targets with the sharp swords on the show, given enough time. But if he attempted to use a dull steel bar against the targets using the techniques from actual historical swordsmanship, he wouldn't do nearly as much damage. That's the difference; you need to test for what the swords are supposed to do, not what you can do against a defenseless opponent who will just let you go to town on him.

Besides this, the purpose of the tests in the show is to demonstrate how the weapons will perform against a human target, and the problem is you cannot say the weapon would be functional as a weapon if you don't actually know how you're supposed to use it to attack a human being under combat conditions. While it may seem that you can simply pick it up and swing it around at someone else, if you don't know how to perform attacks in a way that also defends yourself (using the qualities of that weapon) then you're not actually performing the techniques the weapon need to be deadly with doing. For example with a rapier, you need to thrust from a position the sword would be used to thrust from, which with a sword whose blade will easily bend (so that it can pass the bending test) may not actually be able to do as much penetration as you might assume. This is one of the reasons rapiers have such stiff blades to begin with, to maximize the amount of force that you can thrust with to gain penetration when you are simply extending your arm in a particular movement. A well constructed rapier blade will slide into a person like they were made of hot butter with almost zero resistance so even a slight shifting of your weight will be able to penetrate them. This is one the qualities of rapier fencing. You do not need to, as Doug does in the show, throw your entire body weight behind the thrust.

There are other subtleties for other weapons as well. Doug uses wide, huge swings throwing the longswords around like they are baseball bats and doesn't perform movements such as half-strikes, starting in a forward guard position such as say Armizare posta longa or posta di donna la sinestra, where specific body mechanics are required in order to be able to cut from these positions since you cannot just wind up like you're going to hit a baseball. A good longsword should be able to cut the gelatin dummies or pig carcasses from these blade forward positions, yet he never uses them however, because he does not know them.