When I GM, there is an area of uncertainty between "yes" and "no" where I'll often roll dice. For example, if I don't have a town planned out and if the setting doesn't have an inn in most of the towns off of trading routes (as you mentioned) and so forth but there is a small possibility that there might be an inn where you normally wouldn't expect one, I'll roll dice. If the dice suggest an inn, then I'll figure out why there is an inn where there normally shouldn't be one (e.g., maybe something brings travellers here, maybe it once was on a route and just never closed, etc.).
I roll dice in cases where I don't have a good in-world reason to make a decision either way, and I specifically don't want to make a decision myself.
Combat, for example. It matters whether the arrow hits in one place, or in another 2" down and 3" to the left. I don't usually have specific enough information to say exactly where it hits: diceless resolution would be arbitrary, without sufficient foundation. Since I hate to make an arbitrary decision against a player that might take a character out of action, I'm biased in the players' favor; but over the long term that's bad, because there isn't any in-world explanation for the bias, and it makes the risk perceptibly fake. So, dice.
But I did entirely diceless resolution as the lesser evil until I developed the technique and the mechanics to speed diced resolution up in chat.
And in my experience, where something is strictly possible but very unlikely (e.g., "I randomly shoot my gun into the air over their heads. Does anyone get hit?"), it often makes the GM seem less authoritarian if they roll some dice, even if they set the odds very low, and if the dice turn out to show that the very unlikely even happens, I think that can add a certain sort of realism to the game because strange but true stuff like that do happen in real life.
I don't throw visible dice, and since we're playing in chat the players can't see if I'm rolling or not. This isn't something I'd necessarily carry over into face-to-face if I ever ran that way, though. I have a particular reason for it.
I ran entirely diceless for a long time because all the mechanical systems I'd ever seen had been an intolerable drag in chat. When play is proceeding at the speed of the slowest typist, spending any time at all discussing who's rolling what pulls attention away from the world for much longer than it would face to face; and it gets a lot worse if there are multiple rolls to accomplish a task, particularly if anything is a special case that needs a lookup. So well before the publication of Amber DRPG I'd dropped into dicelessness as an emergency measure. And I didn't start using dice again until I had a simple opposed-roll mechanic with low handling time, and MUSH code to support it so I could get a result by typing at most 13 characters. I silently invoke the dice code when I want it, because discussions about mechanics are way, way, way too slow. They're death to ICness in chat.
If anybody doesn't think a result makes sense, they can tell me about it, whether or not I got there using dice.
As for the character proposing something unorthodox that should work, I think the GM also needs to evaluate the implications on the entire setting. For example, if a player figures out a trivially easy way to avoid being scryed (not necessarily what happened in your example), it raises the question of why other people in the setting haven't discovered it and why everyone isn't using it. At that point, you have to figure out how to reconcile the implausibility in the setting before giving the player an answer.
In this case, the character's far enough above average ability that she can expect to do things most people can't, so it's not a problem. But, yeah, we had a conversation like that, about a spell one of the players was proposing for a background character. If it'd worked the way I at first thought he meant, it would have had major and contrary-to-fact implications in warfare. But on further discussion it turned out that he'd meant it to work in a different manner, so I let it in.
There are some GMs who use a similar plausibility test to adjust encounters. For example, if the PCs find an obvious but easy way to sneak in to the Evil Overlord's castle and kill him, why hasn't anyone else and why hasn't the Evil Overlord found the weakness. Some GMs will then figure out some way to fix the weakness so that it's not there, because they consider it implausible to be there. Again, I think dice can help keep everyone honest and things more realistic. There is always a small chance that the Evil Overlord missed something obvious and maybe nobody ever looked for the obvious before. So before changing the Evil Overlord's defenses, I'd probably roll to see if, rather than being an oversight on the part of the GM, the Evil Overlord really did leave a gaping hole in his defense for the PCs to waltz through.
(Please note that there is a difference between a GM who builds up the Evil Overlord's defenses because a weak spot seems implausible, a GM who builds up the Evil Overlord's defenses because they want the players to be challenged, and a GM who builds up the Evil Overlord's defenses to ensure a particular sort of climactic battle between the Evil Overlord and the PCs in his throne room.)
Well, with a case like this, sometimes the fix is simple and obvious, and then I might just do it.
I might get a player saying, "Hey, why is the Evil Overlord dumb enough to build it like this when people have got to have been using tactic X for generations?"
Or if I'm the first one that notices the problem, I might throw the question on the table. "Hey, I said this, but given these other thing, what I said doesn't actually make any sense." "Well, maybe -- " We all want the world to be believable, and if everyone has had a chance to weigh in with suggestions, it can be easier to get a solution that everyone will buy into. Sometimes I do this when it isn't a problem, exactly, but just an open question about how something works, or how something had to have happened.
I guess you could say that, for technical reasons, I don't throw the dice on the table. But I'm very likely to throw my reasoning on the table.