Well this thread has arrived at pretty much the exact right time for me! Thanks to all for the generous contributions in here which are helping me to figure out how I feel about this.
One thing I've been thinking of a lot is that I can't really get down with games of yore that don't build in any sort of continue, save, warp or password system to allow you to pick up where you left off if you die or run out of continues. It's such a horrible feeling in the modern era to die near the end of a game and have to start all the way back at the beginning. I've NEVER beaten Sonic 2 despite it being one of the like 4 games I owned as a child. I played through it a few months ago and got to the final boss, only to hemorrhage lives and continues and be presented with the title screen because I literally couldn't figure out how to damage the boss. What a horrible feeling! So I want to penalize games that didn't have the foresight or generosity to build in any type of save or unlimited continue system, when it was technically possible to do so. Or rather, I want to give games a boost that deigned to do so. I recently bought the Thunder Force IV port for Switch and am playing it for the first time. Shooters are a blind spot for me. I'm just getting annihilated nearly instantly. I get that there were the economics of an arcade game as a business that needed to be considered when making it, but it's hard for me to laud it just because of how hard it is, no generosity/friendliness for the player, despite the graphics and feeling of movement when you hold UP to soar all the way up to the top of the level being an extreme hell yeah. Even Alien Soldier put in an unlimited continue system. For me, Thunder Force IV can't be a "good" game, no matter how good it looks and feels, because the consideration for the player just isn't there.
At the opposite end of the spectrum from Thunder Force, I've noticed, is like a Final Fantasy VIII. I've been checking out the remaster for Switch. It's insane how badly balanced it is, somehow worse than Final Fantasy VII, random battles where enemies literally do like 18 damage to you when you have 3000 HP. A feeling of utter pointlessness and sense that I'm wasting my time.
So as other people have already said, for a certain type of game, say an "action" game, the challenge is there to give weight to players' moment-to-moment decisions. When the player fails to meet the challenges presented, the designer must decide whether the player gets another chance, and under what circumstances. And traditionally, we see this expressed in a continue or save system. AND, further, I think we can say that assessing the balance between the difficulty and the continue/save system gives us a good view into whether the game is, on the whole, fun. I'm sure to some that's oversimplifying, but that's typically where I start. Games with high difficulty and no opportunity to try again, to me, can't be fun. The payoff for finally succeeding cannot outweigh the frustration of failing over and over. Games with low difficulty and plenty of opportunities to try again can't be fun because there's no weight to the gameplay. The best action games strike the right balance.
There was tremendous weight to that final boss in Sonic 2, I was on the edge of my seat, literally sweating, but when I lost, the feeling was way worse than any sense of triumph I might get from eventually beating it. That's as an adult playing it in 2020. Conversely, to make the easy comparison, I think Mario has consistently gotten the balance mostly right over the years. And I'm saying this as someone who only had a Genesis and didn't play Mario as a child because I didn't have an NES or Super NES. Extra lives in Mario 3 are much more plentiful than in, say, Sonic 1, and there's a warp system to allow you to go right to the levels you're struggling with. Super Mario World's save/continue system is good if kinda inelegant. You just go back a few levels when you run out of lives.
The best "hard game"-type experience I've had is probably Hyper Light Drifter. The actual combat is definitely "hard," in that most players will die over and over throughout the game, but it just deposits you back at the start of the encounter nearly always. The end result is that you begin to pick the encounters apart, like a puzzle. The encounters become way more like a non-game "puzzle," like a Rubik's cube or something, than any Zelda puzzle. This is because you have a specific challenge in front of you, kill all the enemies, and have a limited set of tools to reach the solution -- dash, hit enemy, shoot gun, be able to get hit 3-4 times before dying -- and it's on you to figure out how to best apportion those. So I start to evolve a strategy. OK I'm going to run in the room and unload my pistol on the enemy who immediately starts shooting at me in the corner. Then I'm going dodge because at this point the big plant monster has charged at me. Then I have time to hold down the button for my charge attack and decapitate the three little dudes who come at me. Then it's just me and the big plant monster. As an example. When I die I figure out what didn't work and think about how to better make use of my tools. When I succeed it's because the creative solution I evolved was enough to surmount the challenge. That's satisfying and it's made possible because the right balance has been struck between dying and being able to continue.
However, I think it's possible to ding Hyper Light Drifter because the repeated deaths aren't really explained by the game world. Dark Souls explains repeated deaths with the little story note that your character is "undead" and can never "die" in the traditional sense. That's cool I guess, actually pretty contrived in a vacuum, but what's much more interesting to me is that repeated deaths are one of the mechanics used to deliver the game's story, lore, whatever. To me this is paramount to what makes it such a compelling game. The world in Dark Souls is so bleak, so awful. When you're in The Depths and the horrible butcher enemy wearing a bloodstained bag on their head crushes/slices you in half with their huge blade, you instantly die, because what else would happen? That's lore, that's story, in action. That's amazing to me, even if I don't actually like the experience of playing Dark Souls that much because I'm bad at games and it's just too hard for me. It's fascinating to me that Hidetaka Miyazaki says he didn't really play games growing up because his family couldn't afford them. To me that suggests that the extreme difficulty of Dark Souls isn't a nod to old, hard action games, as it's often portrayed in the gaming press, but merely a desire to make the gameplay consistent with the game world. It's the rational choice to make when designing an extremely dark fantasy world. There's a fundamental incongruence between the tremendous effort modern games often expend via cinematics and dialogue on making an enemy character seem "strong," and being able to defeat that enemy on the first try.
I think Suikoden II really nails difficulty in almost every aspect. Random battles don't present a challenge and are more of a puzzle in that the battle system, which plays extremely fast anyway, offers you various tools to clear the enemy encounter quickly, and it's on you to figure out how best to do that. Most encounters can be cleared within 1-2 rounds if approached properly. It's all successfully explained by the fact that by the midpoint of the game, the story is telling you you're Very Strong, and so are the people you're traveling with. It makes sense that you're able to easily dispatch most enemies. Conversely, the main antagonist of the game is portrayed as being Very Tough, and when you actually face him, he is. You're only able to defeat him by taking him by surprise, forming three separate parties that ambush him each time he tries to run, and then you still have to duel him one-on-one, where despite him having a visible HP bar that is like 2/3 gone, many players still die. When you finally beat him, everyone in the story assumes It's Over. You beat the hard bad guy and it should be. All the characters end up really surprised when it's not, because it's a game about the bullshit reasons wars start and how they affect normal people. That's storytelling to me. That's using the medium of a game, and game design, game difficulty, to do something very powerful with narrative. But I'm digressing :P
I want to say something about Animal Crossing. I've never connected with an AC game despite multiple attempts. For me the various tasks, chopping down trees, fishing, whatever, are too easily reduced to going up to stuff and pressing the action button. The illusion of the game just doesn't work for me. Of course, many, many games can be reduced to just pressing the action button at the right time. That's all we're really doing. But I guess what I'm learning is that, for me personally, the good action games successfully hide that behind building some challenge into that process. I sort of feel lost because I've had to admit as an adult that I'm actually just not very good at games. I can't play competitive FPS games. I can't react fast enough, it takes me forever to memorize control schemes, I probably experience more frustration than most people because it takes me so long to learn.