I’m not saying I’d never do this but I do try to avoid this; if it is just "roll until you get it, no consequences" I generally explicitly tell my players before the roll is made "this is to determine how long you take, not whether you succeed." And if an encounter or portion of an encounter has no failure condition, I just… don’t call for a roll. If the die’s only function is to hamper their progress why am I taking out the dice?
In this instance I might have the statue guarding a shortcut but there’s a longer and more dangerous passage in that they don’t want to take but is a way forward; that way if they fail and have no new ideas for awhile I’ll just say "I don’t think you’re able to find a better way through. Unless someone has a new idea they want to act on right now how about we move on?"
I once ran my players through a fey dungeon that was directly on the border between the Seelie and Unseelie and was half controlled by each faction.
Aside: I don’t really like there being "good fey" and "evil fey" and prefer to divide them along the law/chaos axis. Chaotic neutral Unseelie and Lawful Neutral Seelie. I’d much rather ask the fey have blue and orange morality than have some always be villains and some always be heroes. Anyway!
The front door on the Unseelie half of the dungeon told the story (through a carving) of the birth of the twin queens and ends with their names being called out. The way to open the door is to proclaim your name, though this isn’t any more spelled out than I said.
The players didn’t figure this out, or if anyone suspected they didn’t like the idea of a magic fey door knowing their names and kept quiet about it. They climbed along the roof until they found a gap that dropped them in the dungeon.
To actually answer the OP’s question about the players distrusting phenomenon, I do have a potential way to build trust in players:
At session zero, ask for a "wish list" of scenes or events they want for their characters. Then, early in the game, within the first five sessions, make one of them happen.
Boom, the players have just seen their trust rewarded. And try and include the rest as well, within reason.