I think one of the hardest things to establish is the line between a "plotted" adventure and a railroad adventure.
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At one end of the continuum we have the Complete Sandbox Game
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On the other end of the spectrum is the Absolute Railroad Game
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Is one "better" or "right? No. The only consideration is whether or not everyone at the table is having fun.
I've got nothing against your advocacy of fun. I agree. I don't really agree with your analytical framework, though. It leaves out the sort of game that I like to run, and that I (perhaps being biased as a proponent of it) think can be useful for resolving some of the issues that seem to have come up in the OP's game.
There is an approach to RPGing in which the GM has strong authority over framing the situations (for example, by deciding that an NPC accuses a PC of rape, and by deciding that guards come to arrest that PC). So the game is not a sandbox, because it is the GM rather than the players who has primary control over what it is that the players confront (this can be achieved in any number of ways, from retrospectively motivating NPCs, to determining that whichever door the PCs open, it will be the one with the interesting puzzle behind it).
But the GM does not exercise control over the resolution of the situations. Once framed, situations are resolved in accordance with the action resolution and shared table norms, as driven primarily by the players playing their PCs, but secondarily by the GM pushing back with the NPCs, environment etc.
Depending on the upshot of a situation, new parameters have been established which will determine what is a meaningful, sensible, permissible-within-the-fiction, etc scene for the GM to frame as the next situation for the players to engage via their PCs.
This sort of play is not a sandbox, but not because it heads down the spectrum towards railroading. It's because it allocates clear roles and responsibilities to the players and the GM, and (most importantly) lets the players make the choices they want to make to resolve the scenes that the GM confronts them with. "Plot" emerges out of the actual play of those scenes. It resembles a sandbox in some ways, therefore - there is no predetermined BBEG, for example, because how the PCs will respond to and deal with any given NPC isn't dictated in advance. And there is therefore, and similarly, no predetermined endgame (and so it is the opposite of an adventure path in this respect).
This sort of game will fizzle if the GM is unable to keep coming up with situations that the players find interesting. It will likewise fizzle if the players are afraid to engage those situations via their PCs, and hang back rather than give the game everything they've got. For these reasons, some mechanical systems will suit this sort of game better than others (eg ones which engender trust and tend to make the stakes clear), and likewise some approaches to PC and world building will suit it better than others (eg it helps if PC builds produce clear flags from the players as to what they are interested in doing with their PCs). I
can say with a very high degree of confidence, if I wanted to run a game involving false accusations of rape against a PC, and wanted to minimise the likelihood of the game going off the rails, I would run it in this fashion.