Skills in D&D 5E are very loosely defined. This is a feature, not a bug. It keeps the rules simple and allows for fast play. No skill system, no matter how detailed, can cover all eventualities, so the 5E system doesn’t even try, instead letting the GM make rulings when necessary.
I'd say it's certainly an intentional design choice, but I'm not so sure I'd call it a feature.
The point about the futility of "cover[ing] all eventualities" gets held up routinely as an argument against trying to put an objective skill system in place at all, when a more developed skill system will always provide less need for actual work on the GM's side. The cases that require new rules are sparser and sparser the more descriptive the base rules are and the distance from those rules you may need to wander to derive a resolution in those rarer cases is smaller and easier to bridge.
That, and you don't actually have to use a detailed skill system, even if it does exist. The 5e skill DCs read like a heuristic for a more developed system, and I could absolutely imagine a world where they get used that way, until the one session you decide it's really important to figure out
exactly how many days old this corpse is and go look up the more obscure forensic uses of the Medicine skill.
Simplicity is
expensive. You pay for it with GM mindspace, inconsistency, and in the worst case scenarios, player disempowerment. That, and I'm not actually sure that simplicity is what's actually desired by players, so much as elegance and intuitiveness. In play, it matters a lot less that a wall's climb DC is derived from a description of the wall type and 3 situational modifiers than how accurate a player's intuition about their wall-climbing ability regularly holds up to their attempts to climb things.
Some players, particularly those coming from a 3E background, may be uncomfortable with the lack of detailed skill descriptions in the 5E rules. For my own group's use, I put together a document which aims to bridge the gap between editions, by collecting and presenting some specific rules for skill use in 5E.
This continues to be my single biggest point of frustration with every RPG I've picked up in the last 8 years. Objective skill DCs have been either abandoned in favor of very generic DM advice, with the occasional check called out in a module or odd usage here or there, or in favor of generic resolution systems that aren't fundamentally gameable.
Though to be fair, even if there were a more detailed DC system, I think skilled PC capability in general has suffered in 5e. Skill specialists have to wait until very late levels to push even "Moderate" tasks off of the RNG and suffer worse on opposed rolls, so it's hard to actively use skills to resolve situations.
I had really hoped this was one area that Level Up would lean in harder on, but its skill section is about on par with suggestions in this article. This is good work, and there's several useful examples. I just wish this was the starting point, instead of the best we're going to get.