The part where the 10, 15, and 20 is tied to specific actions.
DC 20 is Hard.
What is considered Hard? Hard for whom? How successful should an expert be at Hard actions? How successful should an amateur be at Hard actions? What is considered being an expert or ameteur?
This seems to me to raise a bigger (or at least more far-reaching) question, which is
what do we understand the DCs to mean?
In Rolemaster, DCs are meant to express something about the fiction. They are a tool the GM uses to establish and convey the setting. Burning Wheel is the same, and explains it more clearly:
BW Gold Revised (pp 72-3): It is the GM's job to set obstacles [= DCs]. By presenting obstacles where he sees fit . . . he builds the mood of the game. . . . it is the GM's job to sculpt, pace and nudge the atmosphere in a certain direction. And not just through beautiful descriptions, he uses the game mechanics to reinforce those descriptions.
BW Codex (p 132): [T]hese obstacles create setting. . . . The obstacle is the number, but it's also the object of adversity in the fiction. Obstacles, over time, create a sense of space and logic in the game world.
Compare this to, say, Robin Laws's HeroQuest Revised, where DCs are a pacing device somewhat independent of the fiction:
HeroQuest Revised (p 74): Resistances [= DCs] are usually assumed to have all complicating or mitigating factors build into them . . . Even when PCs re-encounter a previous obstacle, you can change the resistance directly if the pass/fail cycle [= the game's approach to pacing] or other dramatic or pacing reasons indicate that this is the most entertaining choice.
Make sure that you describe changing conditions so that the change in difficulty appears believable. [Examples are then given: weather for archery; market fluctuations for investing; having hired a new mechanic for vehicle performance; etc.]
Which is 5e meant to be more like? Is DC 20 an objective property of a fictional circumstance, that is meant to also tell us something about (eg) how a typical NPC might experience it? Or is it a choice about how to manage the pacing of play, and so meaningful only in relation to these PCs in this situation?
It seems deliberate that the 5e rules don't answer these questions, as they wish to present themselves as amenable to a wide range of approaches. But any given table might want to form a view about it, to help inform their own approach to the game.