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Pretty much, if you're lacking Sensitivity you won't be able to complete a bunch of subquests, and some of these are necessary to beat the game (not just get True Ending) so you'd just get stuck, really. It's a required resource. The church painting is a good example, because you HAVE to recruit Teulia no matter what. The Orc quest, Funbaba quest, Leon’s statue… the list goes on. Other than that, not much changes. Enemy Search and Secret Search still function normally even with no Sensitivity, but they will warn you that you’re low. All things considered, the dev was pretty careful with preventing Sensitivity-based softlocks during story events, so there aren’t any funny examples where it can screw you over. In fact, there’s even a function where if you have negative Sensitivity at the end of a day, it will reset back to 0.

To clarify, the door itself can be opened after simply getting 100 points for the art gallery and listening to the Ly story, but doing so has no effect on the story until FE2 is completed, upon which you can actually get invited to Lance's house before clearing FE5, but you eventually reach a point where you have to clear it, so yes, I essentially worded the guide in such a way where it means: "Don't open this until beating FE2 and FE5." You won't break anything by doing it earlier, but this way ensures a smooth flow.