that 1 unwinnable fight in the beginning of the game and 1 inpassable check
My opinion on that is this: I am generally against any checks or challenges that player is unable to overcome and serve only some narrative purpose to "teach that player a lesson" by that annoying designer. Now, within roleplaying situation Henry is at that time, I'd say designer is easily within his rights to insert any inpassable checks or fights, due to the fact that you are fucking weak.
However, I believe that regardless of roleplaying situation, a player always has to have an option to do better or worse in any situation at least technically, and see the results.
If they wanted to point out how weak Henry is or to make a particular point, I would much rather prefer the way these things are done in Age of Decadence. AoD actually had fights which you could win but you still ultimately lose. One of those happened in second city right near gates during some main quest I believe - you can get in a fight with guard, you then can fight further with like half a dozen armored guards. It's not an unwinnable fight, but after you win it you basically get a game over screen saying that every guard came and they killed you. Good job.
So to make Henry seem weak an hopeless, fight could proceed say like this: if you land solid hits on bandit, you gain alternative game of thrones scene where he is totally pissed and orders his goons to stab you from five directions with five spears everywhere. You can then plead for your life. If you tell him to fuck off, you die. If you plead, you gain enough time for Theresa to save you and game proceeds.
This way we're doing multiple things - we are allowing player to technically win and show his skill and treat his ego a little bit, but we also break his perception on metagame and narrative and what his character actually is.
Realistically you should always give player a choice even if that choice is to kill himself, because it would still be his choice and death is a realistic consequence for that.
that 1 impossible persuasion check
I am actually not sure all persuasion checks in the game are meant to be passed at particular times or are a viable option. Recently I was at Sasau monastery and tried to make local monastery leader to look like scum and got 1 day jail time for that by failing my speech check.
I think some checks might actually be traps to fool you into believing about the power of I Win Button. Or maybe they are just very high. For example I had to have really high Charisma to pass a check for that hag for Elishka. And then I decided that since that hag is such a total bitch, fuck it, even as I won, I won't leave Elishka at her hands - better to make her stay with Hermann.
It might be again they wanted to break player's perception of being instantly good at anything. But impossible checks are never the elegant solution. Passing check and making situation worse would be a lot more interesting, as long as you prove player with hint that he maybe should take some other option than trying to be all smartass.
I was intimidated by it at first, but after ~10 hours of play I never noticed it anymore. Completely used to it once you figure out timetable of your game and all the ways to save game like rest/baths/quests + savescum potions cease to be any problem to buy or make in any quantities you want. Literally first thing you should do about your character development in any RPG is solving the economy of the game. Once you combine "so dat is a potion" with "alchemy skill" in your character sheet and learn what trader sells what in the game it should be all obvious.
Economy is always the weakest link in any CRPG system. KCD is no exception, although it tries with it's sheer number of money sinks and very intimidating start.