If you are rolling in the open, the players will know immediately that they failed that stealth roll. That's the point you haven't even addressed. I mean if the Vizier is reacting violently to being lied to, then they'd know immediately but there are plenty of cases where somebody who has seen through your bluff isn't going to call you out right there and then. That's the advantage of rolling in secret, the players don't know what's going on, they don't know if they've failed or succeeded at their stealth check, hell if you're dropping dice often enough without them being actual rolls they might not even know when an actual roll is being made. Which is pretty crucial.
Also it allows you to roll for players in circumstances where they might not be aware of things. For example perception checks to notice hidden doors, objects, or creatures; Sense motive checks to discern lying, those are situations where the player's characters might have to make a check, but it's better for them if they don't know that there is a check being made.
The fact that you can't envision a scenario where somebody knows that they're being lied to and doesn't want to immediately reveal that they know is a problem.
You have either never taken keys from a drunk guy, or you really like fighting drunk guys. You definitely should not tell them till the morning. Period. Because what's going to happen is that because drunk people cannot make decisions or think, they're going to want to get those keys.
What if those rules allow for the DM to alter those rules? Most of the games I've discussed do have sections for that, both for the DM setting difficulty classes as necessary or altering challenge numbers or whatever the equivalent in the system is. I'm not fudging the game I'm altering the results of one or two rolls.
Not really. Seriously you are blowing this out of proportion. There is no money on the line, there is no real danger, it's a game. And also as I've said, you probably would not realize that I was fudging rolls, I very rarely do, and if you were playing with me, I wouldn't tell you, because it is already covered under both rule zero and the DM's ability to alter the rules.
No, fudging rolls is a way to deal with capricious fate, which is a thing that you can't always deal with. I've had sessions where I haven't rolled lower than a 15 as a DM, that screws players, every encounter is much harder than it's intended to be, and I've seen multiple 20s rolled in a row. That's something that is sometimes okay, but often is not.
Fudging isn't about negating your choices though, particularly not fudging in an encounter table (which is what was being described upthread). Also most people who fudge die rolls do not do so often. Like I think for me I average around one fudged roll every three or four sessions, maybe. It doesn't come up often.