Since we're already talking about using the skill system, lets assume that there is some doubt about success here...
"Automatic success" is when there isn't, though, and that's determined before DCs are set. If the DM calls for a check, but sets a DC the character can't fail, the /DM has failed/ and DMs should avoid the appearance of fallibility as much as possible, it undermines the level of trust needed from the players.
First up, this is rubbish. It's not how any published adventure works
A published adventure is a tool to help the DM along, setting DCs for him is an exception to the usual mode of play, but the DM needn't abide by it, it's there to aid, not constrain.
nor does it make a lot of sense: you don't assign a different DC to each character who wants to climb a wall based on how you, the DM, feel about their chances. And if you have to... that's a problem. At that point I may as well scrap the entire skill system because it's making my job more difficult, not easier.
Why wouldn't you? A climb might be harder for character than another, irrespective of attribute or training for other reasons - hand-holds might be spaced for a human, but too far apart for a halfling, for instance - or conversely may bear the weight of a halfling easily, but not a human. Training can imply long familiarity, which might make a task trivial (no check), while without training it's still a challenge (roll vs DC).
I covered the degree of failure thing too - it's equivalent to lowering DCs across the board.
It's also just part of narrating failure, and doesn't need numbers attached. The DM can narrate a failure as falling further behind someone you're climbing after, or literally falling off the cliffs of insanity...
… a nice one will even make it clear which is at stake...
It's better, but you still don't see a lot of difference between the worst of the worst and people-with-high-modifiers-but-no-class-abilities-that-guarantee-success
That's BA, it's an intentional design feature. The DM can inject a greater difference by narrating success for the high-bonus characters more often, or narrating failure for the worst of the worst more often, or both. He can also narrate success/failure for them differently. When the incompetent fails, he fails hard and comically from his own ineptitude, when the expert fails, it's a fluke and he recovers quickly.
You're expecting too much from the "system" (there isn't really even a 'skill system' in the sense there was in 3.x or RQII or the like, there's the basic d20 mechanic, and proficiency), and not enough from the DM.