Please hear me out. There is another reason it happens in D&D - because that is what they have been trained to think when playing this game. We are all influenced by past experiences, and if their experiences are they can't do anything, then that is how they will approach the problem.
So why not design the game to train them differently? Why not give them the tools so they CAN do something about the problem? The 2024 Barbarian is already going to have players reacting entirely differently, all because of a single level 2 ability that does nothing more than swap an ability score in a skill and give a little lore for it. But with that alone I can picture people trying things with this "primal power" and trying to find ways to leverage it in unique circumstances.
Not only is the problem fixable, we have an example of it being fixed.
I get that is what you want. But, as I stated earlier, some people think that might create other problems and then ruin it for those people it was working for.
Okay, and? Fixing a thing we know is a problem for a lot of people seems to be worth maybe possibly causing a new problem for other people. Because the thing with "maybes" and "mights"? They aren't guarantees.
So, in story terms, the protagonist in this story had to drink versus an undead. They were the underdog, and failure meant
possible death. And through an act of lady luck, our hero survived.
This is a problem? The DM has a hundred paths they can walk down that doesn't kill your character. Ones that continue the story. I will grant you, as a DM, I do not set up skill challenges that have absolutes that can end the campaign. Maybe yours does. But again, that is a DM problem.
(Side Note: For some reason your story reminds me of Sting's song from Soul Cages, about the protagonist having to drink a magical wine against the king of the ninth world, and if he remained standing, his soul could go free.

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The only reason it was "possible" death is because it was a solo game and we were still in the first two sessions. I didn't believe the threat of death, because neither of us would want to have put in that much time and effort into making this campaign for it to suddenly end so quickly. The presentation WAS death, and likely would have left us in an awkward spot. It wouldn't be the first time a game had a death that suddenly we all realized we didn't want to happen.
But, since you seem to be confused why this situation is being brought up as a problem, let me simplify it for you. There are four buttons in front of you, pressing the wrong one will kill you. It is impossible to determine which button will kill you. When you press a button I will roll a d4, and if I roll an even number, you die.
That... is a bad challenge. Because it isn't a challenge. It is literally all about luck. Like you said, my character survived because of luck. Not because of anything I did, not because of any action I took, and in fact there was no action I could have taken other than being lucky.
And I bring this up, because this is COMMON for martials. This happens all the time to them. They have an incredibly limited ability to seek alternative solutions. They lack tools.
You're right, I did use the DM argument. But to be fair, I also asked you for an exact situation that the game put you in as DM that needed to be fixed. You came back with a character example, and even then, I can't see what needs to be fixed from a ruleset standpoint.
No, I gave the answer, then gave a specific memorable moment when that happened to me as a DM (the barbarian). And then a second where it happened in a different way (the monk). Both dismissed because they involved players.
If you will only accept my answer if I give a moment that DOESN'T involve a player, I'd ask how serious you are about discussing DnD. To reiterate "The fact that, consistently, martial characters feel like they cannot participate. Even when I specifically design encounters to highlight their strengths... the casters dominate." If I need to give an example that doesn't involve any players, or any characters, I will try, but I don't see how that does me any good since it will immediately be called a white room and useless.
Again, I have not dismissed anyone. I have asked for examples. I have received very few. Most just say this martial/wizard thing is a problem, so let's make the martial stronger. That, in my opinion, just sounds like half the arguments on these forums: I want my character to be able to do this, and they can't. Therefore, we should change the rules/add powers/etc.
Really? Because the Barbarian situation I ran into was "a player problem" and "just how we are trained" and you seem to have completely forgotten it. Even though it was the first example I gave of exactly the type of situation I am put in as a DM.
And yes, it turns out when characters can't do things to affect the story in meaningful ways, the solution is to give them that ability. Do you expect a different solution? If the problem is that my brakes on my car won't engage, I don't solve it by putting gas in the car, I solve it by replacing the brakes with working brakes.
The problem is martials don't have the tools to compete, so the solution is to give them those tools. That shouldn't somehow invalidate the solution.
We must not play the same game. Because the androsphynx would present such a challenge that the wizard might have a 50/50 chance. Not good odds for someone who has spent years playing a character to get them up to this level. I should add, that is without any DM shenanigans as well. Throw in those, and the odds shift to 90/10 in favor of the androsphynx.
Okay, statted up a quick Evoker Wizard, level 20. Gave them some appropriate magical items for a level 20 character. They would have 122 hp, and an effective AC of 27 assuming shield.
With a +3 dex vs a +0 dex, the Wizard likely wins initiative. Their spell save DC is 21.
They cast Time Stop, take an additional three turns. You might imagine that the lair actions would prevent this, but they do not. The Lair action has to be taken by the sphinx, and while time stopped they cannot take actions.
Turn 1 -> Delayed fireball blast 8th level
Turn 2 -> Fire Shield 4th level
Turn 3 -> Disintegrate (unleash fireball at end of turn)
They have a 70% chance of failing any dex save. I'm estimating 57 damage from the delayed fireball and 86 from the disintegration. That is 143, leaving them with 56 hp.
If they roar on their turn? Okay, I'm frightened, who cares. Next turn I do a 5th level empowered fireball and potentail deal 60 damage, killing them. They use dispel magic on the Fire Shield? Same thing. They attack with their claws? Even if both hit they can't kill me and I finish them off with the fireball, likely a weaker one since they took 4d8+10 damage from attacking me. And they have a less than 50% chance of hitting anyways. Goes to a different plane of existence? Who cares. Moves me into the future? Who cares.
So, no, really. How do you imagine this is a 50/50 fight AT ALL?
To prove I am not being dismissive, I have laid out options (maybe three) on how to balance the game better for those that see a problem. That's with me not even seeing the problem. I have talked about tiering the system and its flavor text (I think words, even flavor words, matter). I have suggested that martials not be bound by the 20 cap on abilities and allow them to raise to 30, while casters remain at 18 or 20. And another way to solve the issue is to limit spells by adhering to components, finding the spell, or simply removing the one or two that seem "breaking."
I have proposed all these things in this discussion. None of them work for people because they want their martials to have more. It's a never ending cycle.
Sure, changing the flavor text is a start, but it is rather meaningless without the tools added. But hey, I'd love for Martials to not get the short end of the flavor text. Currently mages explore the multiverse, druids become one with the planet, clerics become the right-hand agents of their gods.... and martials get a family and a castle. Not exactly the same things.
Not being bound to 20 as a cap is a good idea.... except that there are almost no ways to raise their ability scores anyways. Sure, a fighter who starts with 16 strength could use every single ASI they get (they only get 7) to get a 30 strength, but they get no feats, no other scores increased, nothing. If there were easier ways to raise ASI's and still get feats which are VITAL to giving martials options, then this could work, but you have to redesign how ASIs work in totality.
And the spellcasting thing is a red herring. Adhering to components is 95% a waste of time, because the focus is the compoment, only in VERY rare situations is that not true. Having to find spells could work, but that would only affect bards and wizards really, and frankly could encourage DMs to give them more than their 2 spells per level anyways. Yes, we need to change the worst spell offenders, but that still won't do anything to help martials when they can be out-done by the normal spells too. No one says Web is a broken spell, but it sure can fundamentally alter even a high level encounter in a way that martial characters are simply incapable of.
This isn't a never ending cycle, because you NEVER gave martials any new tools. Your best suggestion was bigger numbers if they give up all their feats, maybe. People are saying they want a new engine in their car, and you are throwing your hands up because you offered a new paint job, new wheels, and poured sugar in the neighbor's gas tank and they still aren't happy.