The game text doesn't present skill usage as a binary pass/fail... not even if you stop at PHB 174.
And yet... in practice that is seemingly how they are treated in all other situations in D&D when they come up. I mean, can anyone find me an official adventure where failing a DC-based skill check isn't treated as failure, but "progress with a setback"? I've certainly never seen that, but maybe it's out there.
Among those rules is DMG 242 which gives the method for non-binary outcomes. The range of results from ability checks is broader than that even, including (this is all from non-optional game text)
Can you give me a heading or something? I have the DMG on Beyond, rather than physical, and page references don't work, but I can't find anything like this. Not saying it isn't there, just that I can't find it.
EDIT - Do you mean the section "Resolution and consequences"?
That's an interesting one because it's somewhat at odds with the PHB, in that it doesn't even include the "progress with a setback" option (rather instead having "success at a cost", which is almost the inverse, or at least 90 degrees off), among other things.
It's also misleading, I feel, for you to lay things out so neatly and clearly, when the book completely and totally fails to do so. I had to look for a lot longer because I was expecting something clear, but in fact what you've got there is a much better and more useful clarification than that the DMG actually offers, which is vague text waffle.
I definitely hope 6E put this sort of thing in the PHB, it's genuinely non-optional as you say, and makes it a lot clearer.
Also, you say it's "non-optional text", but the text is literally phrased as if it's "at the DM's whim", so I mean, it's obviously optional just without an "optional rule" heading or sidebar.