What's the DC to dethrone an illegitimate monarch?
I'm not going to entertain that first point, because we both know it's ridiculous and I'm already tired of having my position caricatured. DCs are for discreet actions, not geopolitical goals.
What's the DC, even, to cross a river?
The DC to cross is a river, in the kind of skill system I'm proposing, will be resolved by the appropriate swimming, levitating, flying, rope tying, boating, long jumping, or whatever other rules the PCs bring to bear against the challenge, no doubt informed by the speed of the current, presence of monsters, visibility and anything else going on.
Unlike a skill challenge, the number of times they roll will vary wildly, based on which of those approaches they took, the appropriate speeds of the methods of transport available to them as a result of those choices, and in some cases will be trivial, as perhaps someone actually bought a
folding boat. Honestly, the only difference between the sort of system I'm advocating and a skill challenge model, is that DCs are intrinsic to the tasks being attempted and the effectiveness of any given check is specified by the action that allowed the check in the first place, instead of attached to a timer on the number of checks or number of successful checks.
I don't think it's irrelevant to this discussion that the concrete example you give of constraining fiction is a wall that the players might want their PCs to circumvent. Even then, most discrete action resolution systems that I'm familiar mechanise the smashing through of a wall by giving it "structure points" and resolving actions analogously to D&D attack, the aim being to wear down those points.
Oh totally. Give me structure HP, and we can figure out if there's a downside to attacking the wall, or if climbing is faster/safer, or if some completely unrelated approach will most efficiently solve the problem. The way I play these games is essentially iterated heist planning.
So, sure, maybe the PCs come up with a power or something that gets them out, but chances are you can still provide some checks to see if it is feasible (IE can they perform the ritual 'Dimension Door' while blocks of stone crash down around them).
But I disagree as to what you are 'better off' doing. Think of it this way, if you specify a bunch of specific obstacles, jump the cravasse, dig out the collapse, find the air shaft, etc.) now what if the PCs decide to sit and do a teleport ritual instead? Your prepared 'stuff' is no good, you can obviously come up with other dangers, as I mentioned above, but so can I in an SC!!!! I don't see how that structure, in a mechanical sense, constrained me. We both want obstacles to overcome. It is really no different, I just have a measured number of them that I will deploy according to a rule.
This is the fundamental difference here. In a system with objective actions, players are incentivized and rewarded for trying to find the most efficient solution to that problem, and will go off to deal with new problems, while in a skill challenge scenario they cannot have made a good decision about which path to take, because you specified a set of obstacles ahead of time. In some cases you might be able to resolve a situation thus that you face no or fewer obstacles than some other path, and the satisfaction is in finding and using that route, an option that isn't possible with a skill challenge structure.
Certainly the skill challenge will produce a consistently cinematic set of events you could narrate later, but it isn't interesting as a game to play in the moment because you cannot do well as a player. The entire proposition is to use a skill challenges as a structure to generate fiction that's sufficiently exciting to talk about, when that's precisely the opposite of what I want to do at a gaming table. I want to receive a fictional situation, figure out the best way to resolve it to my advantage, and be rewarded if I manage to do so.
I don't think it's controversial to point out that this is something skill challenges don't and by their structure, cannot, do. That they do not do this is straight up getting called a strength in these examples, as they point out how you might set up a new challenge when you check the fictional situation before that last success is rolled. My point is that this strips them of a particular kind of gameplay.
To summarize, in a skill challenge stripped of its fictional context you roll dice and try to get high numbers, and every once in a while you can cut down the number of times you need to do so by a third by spending some other resource. If the nature of the challenge was allowed to scale on the number of relevant rolls and the difficulty of those rolls was unbounded from the challenge, you would have more tactical engagement to make better choices. At that level of mechanical engagement, a skill challenge system loses out to a specified action system in terms of player agency.
It's very possible not to care about that. To be honest, I actually think a lot of these defenses of skill challenges could be restated as claims that lower mechanical agency makes for more consistent pacing and story structure, and allows for more freedom in narrative agency to players. That is a reasonable trade-off, one could in good conscience support, and specifically write mechanics for it, and arguably a bunch of indie games have. It's just not as interesting to play as a game, in the very specific sense I've been using the word game.
If that's what you want from your non-combat system, that's completely reasonable, but it's not an intrinsic good and does sit in opposition to various kinds of fun you can wring out of a roleplaying game.