Sure, that can happen in any game, but the answer isn’t for the DM to do a lot of extra work. It’s for the table to pause the game and have a discussion about what’s going on in the fiction, so everyone’s on something like the same page, because until they are, the game cannot proceed functionally. Once that has taken place, the player can make an action declaration that makes sense within the context of what the table agrees is the established fiction. I wouldn’t consider this process part of gameplay per se but rather a reestablishment of that part of the social contract that governs “this is what we’ve all agreed to imagine together”. It’s also why I don’t think examples of such breakdowns in the social contract are a valid criticism of rules-text like the background features, because you need an intact social contract before you can proceed to play the game. It isn’t the features’ fault if participants in your or my game aren’t on the same page about what the established fiction is.
I'm not sure what you mean by "narrative game", but from what you're saying here it seems you mean a game in which players are allowed to declare actions for their characters that play some part in "determining the world". It's simply not true that this isn't a "default assumption for the game", assuming you mean D&D. In a game of D&D, if I as a player declare an action for my PC to attack the orc with my PC's sword with the goal of making the orc dead, and the action succeeds, it is my action declaration that determined the world now contains a dead orc. You also state the player (and not the DM) is "100% responsible" for what their PC thinks. So why, when the player says their PC thinks a ship with which they're familiar (the Comox) frequents this harbor, are you, the DM, telling them they're mistaken? At the very least, this means the player is playing a character with faulty recollections, which is probably not the player's intention, and, even worse, implies the DM is more of an authority about what the PC thinks than the player, which steps all over the notion the player is responsible for that aspect of roleplaying their character. It's not surprising, from your earlier posts, that you seem to favor scenarios in which the PCs are in completely unfamiliar territory, as this would seem to support the players playing their PCs as people who don't know anything, which I think is the natural outcome of having to ask the DM if it's okay for your PC to know something. Lastly, I think it's a bit ironic, in a thread where I believe you've accused me of exhibiting one-true-way-ism, that you're telling me that "the default assumption" and "standard approach" for D&D has never supported players contributing to the fiction outside of merely animating their characters. I mean, the background features were published in 2014, so at least at that time it was the default, wasn't it?