I entirely agree with you with regards to the "stakes" - there should always be stakes involved in combat, otherwise it's all fairly pointless isn't it? But, just like in movies, having gratuitious battles just for the sake of having them is kind of silly as well: shouldn't the stakes be something else (e.g. the fate of the kingdom or the girl, to use generic tropes). In other words, why aren't you talking about the stakes outside the immediate lives/equipment/the whole party? It was some time since I played rather than GM'd, but I do know that my players have found it interesting to follow a battle their characters aren't directly involved in, but that nonetheless is significant for the story.
Anyway, regarding the visceral parts: I agree, partially, but I also disagree (or, rather, I think you are being inconsistent again). Earlier you wrote about how important it is for the GM never to fudge the dice or to create his own rules on the spot that aren't followed through - all in an effort to increase the player's sense of agency. But particularly visceral descriptions come from nowhere but your own mind, do they not? So say, for instance, that your player scores a critical hit on a person. Now, you have to figure out how to describe this, and you do: "You swing the metal pipe in a wide arc, hitting the man across the face, causing his jaw to dislocate and his teeth to fly out in a bloody splurt". Whoa, cool, everyone sits up. But the dude isn't dead, he's just badly wounded: he still has HP or the equivalent thereof left. But it doesn't make any sense any more for said dude to be up and fighting, EVEN IF the game mechanics and his hitpoints suggests that he should be!
What I do then is I describe how the guy collapses in a heap, whimpering pitifully as he tries to vainly collect his teeth from the ground or whatever. I'm sure most people would do the same. But this is fudging, this is GM fiat, this is cheating: according to the rules, the critter is still supposed to be up and about, giving your party a hard time. However, the only VIABLE alernative to describing the critical hit in such a visceral fashion is to downplay it ("You hit the man over the head, and he staggers back, momentarily stunned"), which basically moots your entire point about visceral descriptions.
This can go on to the "almost dead" states most games have: you just described the gutting of the main villain, but moments later (as he lies in his 'dying' state), the players decide to spare him (for whatever reason) - how in the earths are you going to let them do that, even if (once again) the sacrosanct rules would definitely allow for this? Of course, this is different in a high-magic/high tech setting, but not all settings are such.
Essentially, what this all boils down to is that what you as a GM are doing when you describe combat 'viscerally' is you are taking AWAY from player agency: you are not just interpreting the dice roll, you are playing their characters. At best, they're like those little insta-death cutscenes in Assassin's Creed whenever you manage to click your buttons right during a fight: awesome cool. At worst, you just made the player do something their characters wouldn't do, you just took over their whole being (such as your rather poor example of the warrior jumping through the air and headstabbing enemies left and right).
I will concede that the problem here, as you rightfully point out in the article, is that in most tabletop games, critters take more than one hit to go down - it can be a little annoying to attempt to describe the plethora of stabs and cuts and (worse still) gunshots something can take before finally going down. Once again, it is like the suspension of disbelief that happens in a video game where the main character is suddenly threatened by a lone girl holding a pistol, despite just having battled through a horde of minigun-wielding super-cyborgs and their rocket-launching giant mutant badger pets. You made a very good point that, whenever it's not a critical or a killing blow, one can "focus on damage to the target's armor and shield, its state of pain or fatigue, and its position" - although this still leaves the problem of both the critical and the killing blow as described above (especially considering the 'pain' of a target creature after a massive critical hit should have real in-game ramifications, even though the rules might say nothing about them - e.g. you cut off somebody's arm).
The alternative? Well, the first part would be to allow for fudging and GM fiat again: rules are meant to be broken for dramatic effect. There, simple as that. Meaning your gut-spilled main villain will die on the operating table before divulging any secrets and your still fighting-capable-but-now-toothless goon will be taken out with a swift kick to the head when the time comes. You have made a decision as a GM, for the sole purposes of furthering the plot and the enjoyment of your fellow players, and this required you to change the rules of combat and healing: so be it.
To lessen the blow of this betrayal of principles, I tend to ask the players to describe what they are doing before they do it, instead of the opposite. This is not something you discuss in the article, which I found rather strange, but then I rarely play D20-styled games which focus heavily on the combat (where I assume it'd get a little boring after a while to keep telling the GM 'how' one swings ones broadsword). But even then, it should be possible. Essentially, just like with -any other action- in the game, ask the player to describe what they are trying to do before rolling the dice, and then interpret the roll as it comes. If they said "I swing my metal pipe at the man's head!" then describing a critical as "You knock his jaw and teeth out" will be met with quite a different kind of elation, not to mention removing a lot of the GM fudging (since it was a result of the player's actions).
This is not to say I don't enjoy showering my players with unintended gore as often as I can, but then again I don't mind the GM fiat it involves - rules are meant to be fudged.
A nice article, as per usual!