My stance is to let the player's decisions matter, because while I'm the arbiter of the world, we're doing collaborative storytelling. When a writer tells you a character does something, that action should matter - to mood, to characterization, to plot, or something.
Likewise when a player says they search a room, I'll maybe ask for clarification to add some texture to the scene, but if they just say they make a search check, I'll use my power as the narrator to guide them to the in-narrative actions that will achieve their implied goal. They want to find hidden things, so if there's a hidden compartment behind a moose head, and they've said their searching the room, I'll highlight things to draw their attention to it. Then they'll narrate specific actions to investigate the head, which then makes any discovery they have feel like their choices produced it.
The dice are . . . not that important. Honestly, out of combat, I weight dice maybe 10%, and weight player intention like 70%. The final 20% is the surrounding context - did they make a character to be good at this, is it going to affect the fun of the rest of the players, etc.
Like, in speech act theory, there is locution (what is said), illocution (the intent of the words), and perlocution (the result of the words). The classic example is "Is there any salt?" The locution is a question about the presence of salt. The illocution is a request for someone who has salt to hand it over. The perlocution is that the speaker gets salt.
I feel like a major part of my role as narrator is to understand the illocution, and ensure that the locution leads to the desired perlocution.