So maybe you're using terms that are just going over my head, but I disagree that the DM must make rules for when to let the players roll dice. The books are quite clear on this and give a fair amount of guidance (even if I hope the 2024 edition adds and clarifies). There are different options presented, but it basically comes down to dice are rolled when there's uncertainty. There are also guidelines on what the DC should be based on difficulty.
I will agree that the DM has to make judgement calls on difficulty of achieving a task that varies from automatic to dang near impossible. But that's not "inventing" anything. This is a game of imagination and make believe, the only reality is the reality that the DM and players make so someone has to make a judgement call. I'm always reminded of D&D 3.5 and it's climb wall DC chart that gave difficulties for different types of wall. But the thing is, the DM still had to decide what kind of wall it was to use the chart, it just offered the illusion of impartiality.
Sometimes I feel like people are talking about a different game then the one I play.
I dunno. I agree with the you when you say "the DM still had to decide what kind of wall it was to use the chart, it just offered the illusion of impartiality." and I've always felt that this type of game, where the GM is arbiter of a set of rules, and a set of fictional situations, where the mechanics and the situations are all open-ended is inherently like that, the GM is the game. I agree that GMs then limit themselves, or the players force them to limit themselves perhaps so that some sort of principled play happens (or not...). I do have a pretty good sympathy for
@loverdrive's position though, that 5e D&D is so loose in terms of what it prescribes, that you practically have to write the game.
Like I'm looking at the DMG P236 where it talks about 'The Role of Dice', and its like, the game punts! I mean, it describes some things you could do with dice, but there's no actual process of play there! Does the author of 5e actually want you to take the 'Middle Path'? Maybe... but basically that's no criteria at all! Now, right after that another section tries to rectify that, sort of... But all it really says is "make them roll if they might fail", but how does that even mesh with the previous section on the Role of the Dice? Now it talks about whether you can roll again if you fail, and guess what, it doesn't answer the question! In fact nothing actually makes clear what a check resolves, intent or action, nor how it relates back to fiction.
This is all before we even got to difficulty, which is on P238, supplemented by various added sections on P239 (and I never could understand why the DMG talks about the GM granting advantage/disadvantage when they already have the set-a-DC lever).
There is after this a section that seems to be basically optional rules for degrees of success/failure, but fundamentally says "When a check equals or exceeds the DC it succeeds", but what does success mean? Failure isn't really discussed here at all, either except in noting that you could make bad things happen if the check result is really low.
I mean, is this actually a game design? I would say it is game designer notes! I don't want to sound too harsh, but its very wishy washy. The first section of the PHB doesn't really elaborate either. In fact it doesn't even commit to the GM ever needing to consult the rules at all! The Section How to Play on P6 literally does not mention dice, except once in passing as something the GM might possibly use! I could literally follow the rules of 5e to the letter and never pick up a single die or ask a player to do so, and I would be running the game verbatim as described on P6, and consistent with the DMG section on running a game. I admit, it would be very odd, but its actually a VERY ODD GAME. I contrast this with Dungeon World, because the contrast is so stark. Chapter 1 & 2 of DW, and then supplemented by the start of Chapter 13, COMPLETELY defines a game structure just as surely as the rules of Monopoly describes how you play it. AFAICT 5e D&D never actually does this. It never clearly articulates the purpose of making checks, nor does it articulate a rationale and thus requirements for the GM's authoring of fiction.
So, the observation that the GM can set anything to anything, and thus all the 'rules' of 5e are an 'illusion of impartiality' is true, but even more profound is that there's no actual process described for how or when to even invoke those rules! Meanwhile DW is utterly impartial; 'difficulty' isn't even really a concept, and the game very carefully spells out when and how every rule works! Sure, there are some judgment calls about "is this possible?" but there's no partiality there, the table decides, as a whole! And in practice those situations don't intrude on play much, everyone knows halflings cannot simply leap into the air and fly away...