Player agency means the ability to affect and change the game world. To make meaningful choices.
I don't think those two things are equivalent, and I think the first is not sufficient for the second.
In almost any RPGing, a player action declaration for their PC will affect and change the game world. It means that the world now contains a character doing, or at least attempting, whatever the player declared. And often it will also prompt the GM to narrate something to the player about what happens - perhaps by reading from their notes, perhaps by extrapolating from their notes, perhaps by just making something up.
But if the player was just, in effect, offering blind prompts to the GM, then I don't think the player acted with much agency or made very meaningful choices.
For me the classic critique of blind prompts, from the perspective of player agency, is found in Lewis Pulsipher's essays in late-70s/early-80s White Dwarf. His focus was on what he called "lottery D&D" - where the players have their PCs draw from Decks of Many Things, pull levers, drink from wells in the dungeon, etc, and the GM reads from their notes or rolls on their chart to tell the players what happens. He contrasted this with what he called "wargame-style D&D", which is more or less Gygax/Moldvay-style dungeon-crawling, based around the players gathering information about a somewhat static GM-authored situation, and then acting in a planned, reasoned way on that information.
I think Pulsipher's critique of lottery D&D generalises to play where the
flavour/
colour of the play is less gonzo than drinking from a magic well, but the
structure of play is the same. And a situation in which the players know that some indeterminate badness will happen if they don't get from A to B in time; that one path to B is short but dangerous; and that the the other path is longer but safer; has the underlying structure of "lottery D&D". The players have no real choice but to declare their PCs leave A for B; and they can either pull the "maybe this will help us" lever, of taking the short route, or they can leave the lever unpulled and find out what the GM's default narration is by taking the long route. It's all very colourful, and in some fashion it "affects and changes the game world", just like pulling levers in lottery D&D does; but the choices don't seem meaningful to me beyond being gambles. And I don't see much player agency.
Often during a game players are working with incomplete information, sometimes without any at all. Given the choice of a T intersection in a dungeon with nothing to distinguish between the alternatives, the choice between them is arbitrary, not informed, right?
This relates to
@FrogReaver's point about time horizon, I think.
If the assumption, at the table, is that the PCs will only traverse these tunnels once, then the choice is as you say arbitrary. In my own play - of 4e D&D and Burning Wheel - in these sorts of situations I typically don't bother working with detailed maps and architecturally-specific action declarations. I use abstract resolution - eg a skill challenge in 4e - to determine whether the PCs get to the other end, and what happens (and what costs are incurred) along the way.
But if the assumption - as in classic dungeon-crawling play - is that the players will traverse the tunnels multiple times, then this is information gathering. And so while, on it's own, it is not all that agential (unless the players have some other information that motivates them to scope out to the left before scoping out to the right), it feeds into the agential dimensions of play. Again, for me Lewis Pulsipher is one of the best authors on this, in those 40-something year old essays.
And if there's any info at all, that's still agency because it's something to go on, some data on which to base a meaningful decision. In the specific example you gave (one road is half the travel time but double the chance of encounters) the difference is relatively small all other factors being equal, but if one is expected to take a week and get them there in time for the coronation in nine days (say) and the other is expected to take two weeks... well, just knowing that is enough to make the fast road a meaningfully superior choice if their priority is getting there in time for the coronation.
This would be a simple application of a rational decision rule. One way of thinking about player agency is that the GM won't retrospectively muck about with the fiction (especially via manipulation of hidden backstory) so as to negate the optimality of these sorts of decisions.
At the table, one way of honouring that might be to roll your random encounter dice in the open.
If, for example, they have the option to turn back, or to spend resources on an Augury spell, or send a stealthy scout ahead down one or both passages to GAIN further info, they certainly still have some agency!
Going back to your random tables of wilderness encounters, I would say that without any prior info about the potential contents of the tables, agency is again relatively low. But especially if they have ways to GAIN further info, such as by seeking out rumors, asking local merchants or caravan guards who travel those roads, then again, agency exists. In such a case, I would be likely to give the players some info about the random tables, either out of game or diegetically. Say the players find and question some caravan guards about the more dangerous route. And I know that the three most common results on the table (6,7,8 on a 2d6 table, for example) are organized Orc raiding parties, small groups of trolls, and a band of Ogre brigards, the guards could tell the PCs about these relatively common threats. As well as maybe mentioning that a black dragon has been seen occasionally (it's a 12 on the table, but it's scary enough that everyone remembers hearing about sightings). Knowing about these threats the PCs could choose to keep Sleep and Fireball prepared, and maybe check town for potions or scrolls of Acid Resistance just in case.
If it costs nothing to get more information but time spent at the table declaring the appropriate actions, then there's a risk of asking players to trade off agency against boredom. I have played in games that had this character; I don't think they're very satisfying.
I prefer to embed the information gathering into some sort of cost/benefit matrix - my favourite game for this, at least at the moment, is Torchbearer: doing research during Town Phase adds to lifestyle cost (which is a downside) but (i) allows you to make a test (generally on Circles or Scholar) which is an upside because it feeds into advancement, and (ii) may give you useful info. Eg in my game the players (via these sorts of declared actions) learned that Celedhring had entered the caves years ago but never left, and so they thought "undead" and prepared themselves by buying holy water, and this helped them when, in the caves, they encountered undead.
The optimisation aspect in Torchbearer is probably not solvable in practical terms, as the various payoffs (risk of failing Lifestyle test; advancement benefits of making the research test; advantage gained by having holy water when fighting the undead) are hard to calculate and also hard to commensurate. But the players are exercising meaningful control over the "shape" and content of the shared fiction.
