The issue is that you keep overlooking the actually important part: how the decisions are made. The GM deciding things is not arbitrary. They're doing so based on pre-established information and the the actions of the characters. Thus the players' decisions guide the GM's decisions, greatly influencing the end result.
I've never asserted that the GM deciding things is, or must tend to be, arbitrary. That the GM decides things for a reason doesn't show that anyone else has agency, though. That one of the reasons is a prompt provided by another person doesn't show that that other person has agency either.
The point can be made in the abstract: the decision to play a lottery prize to A rather than B isn't arbitrary, but A and B don't have agency in playing a lottery.
The point can be made in the RPG context too: the outcome that
@Lanefan has described about the civil war, that I quoted just upthread, is not arbitrary. But the players have not exercised agency in bringing it about - it's all being decided by the GM based on considerations that the players don't have access to, haven't sought to draw upon or shape, and had no say in establishing.
Can you explain how these things are different between a dungeon and a wilderness area or a city?
A dungeon is an extremely artificial environment, invented to be a venue for game play. The paths are learnable by making the moves the game allows for: "I look down the left corridor"; "I tap the floor with my 10' pole"; etc. The scenes/situations that are latent in the dungeon are segregated (in rooms) and set behind player-controlled "triggers" (opening the doors to those rooms). And those situations can be ascertained without triggering them (eg by listening at doors, by using detection magic, etc).
You can see all this in examples (like Gygax's example in his DMG, or the early TSR modules), infer it from stories (like the stories of freeing Fraz Urb'luu or the demigods in Castle Greyhawk) as well as the list of tricks in the DMG (many of which are about making it harder to map accurately), and it is the underlying basis for Gygax's advice in his PHB. That advice is that players should engage the dungeon in various "modes": first, scout out a section of the dungeon to establish what is there and identify the target loot; then, after retreating and changing spell and gear loadout as appropriate, raid the target. This advice makes
zero sense if the fact of scouting and mapping the dungeon also changes it, so that the players can no longer withdraw and then choose to re-enter so as to trigger the (latent) situations they have identified (in the fiction, this means the PCs are exploring the rooms they have found via their scouting). If every foray into the dungeon means starting from scratch in terms of knowledge of threats and opportunities, the advice is pointless.
Another author who provides advice to GMs which reflects this same approach to dungeons is Lewis Pulsipher, writing in White Dwarf in the late 70s and early 80s. Pulsipher in particular emphasises the importance of detection magic, as a tool that good players will use and hence that the GM must have regard to in their adjudication of the players' play.
A wilderness and a city are not knowable in the same way that a dungeon is. The pathways are (for practical purposes) infinite. The threats and opportunities are not static behind doors in the same fashion. And in D&D, this is even reflected in the ranges of detection spells, which are meaningful in the artificial dungeon environment built around 5' or 10' squares on A4-or-thereabouts graph paper, but are virtually useless in the sorts of scales that operate in the wilderness and even in urban areas. Hence wildernesses, and cities, are not amenable in the same fashion as are dungeons to the explore-retreat-raid model of play.
Of course it is
possible to keep some of the basic tropes of a dungeon but to try and reduce its artificiality - a working castle would be an example of this. Gygax heads in this direction in his DMG (which is written later than the PHB, at a time when the whole approach to RPGing is unfolding and changing rapidly). But a GM who follows Gygax's advice about "living, breathing, responsive" dungeons in his DMG is making the advice to players published a year earlier more-or-less useless.
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In game terms, you can't have all three of (i) the GM establishes more-or-less all the content, and (ii) the GM controls the way this is revealed to the players and engaged with by them, and (iii) the players exert significant agency over the content of the shared fiction.
The classic dungeon crawling of Gygax's PHB puts constraints on (i) - the artificial dungeon - and hence largely drops (ii) - the players, by choosing how to engage the content (in the fiction, by having their PCs collect knowledge of the dungeon and then raiding this bit rather than that bit) - and hence achieves a version of (iii): play is about beating the dungeon by solving its puzzles (understood in the broadest sense) and thus extracting its loot.
A game like Apocalypse World or Burning Wheel puts completely different constraints on (i) - the players, by establishing relationships, goals, etc for their PCs, generate many of the salient elements of the shared fiction - and also different constraints on (ii) - the GM reveals the content in a way that puts the player-authored concerns at stake - and hence achieves a version of (iii): play is about the players engaging in and perhaps resolving (in whatever fashion) all those conflicts and emotional/thematic/relational potentialities that are suggested by those PC elements they have introduced into the shared fiction and that unfold further through play.
AW and BW don't need the artificial dungeon as an environment, in order to underpin player agency, because they take a completely different approach from map-and-key dungeon exploration to how declared actions are resolved. The constraints on the GM operate in a different fashion, and at a different point in the process of play.
