Fanaelialae
Legend
The player is not telling the DM that his character knows something the DM thought he didn't; the player is changing the world as the DM knows it.
Was there a shorter route? No. Is there a shorter route now? Yes.
Should that shorter route continue to exist in the future? Yes.
Could that shorter route have had an impact on campaign play in the past? Potentially.
Who has to track these effects? The DM. Now let's suppose that the PCs are engaged in a friendly rivalry to catch the villain that the DM thought was fleeing by the most direct route.
Player 1: Do I know a better route? I roll a 21!
DM: Yes, you'll catch him at...
Player 2: Wait a minute! I know this city better than him I roll a 25!
DM: OK you know an even better route. You'll catch him at...
Player 3: Hey wait a minute! Remember 13 sessions ago when I was fleeing from the ragamuffins! I found a terrific route out of here, I rolled somewhere in the mid-30s!.
How many better routes can exist? How long do they continue to exist? Why didn't the villain traveling by the most direct route take them?
I don't see it that way at all.
Was there a shorter route? Not on the map.
Is there a shorter route now? Still not on the map, but if the check is high enough why shouldn't the player know a secret shortcut?
In my opinion, it isn't even really about letting players run the narrative. It's about using the dice the way the dice have always been used; to resolve an uncertain situation.
If there is literally no logical means for the players to reach the villain's destination before he did, I'd just say no. The fastest possible route is the fastest. However, unless I'm detailing a simplistic cavern or some such, I probably don't have the means to make such a determination without being arbitrary.
A map is static, and only good within a limited scope of detail. Perhaps the city streets are particularly crowded this day, or there's been a carriage accident on that road, and the PC knows a rooftop or sewer route that allows them to bypass traffic. If the villian's path includes turns, perhaps the PC knows a route cutting through shops and private residences that allows them to travel in a straight line. I'd probably require additional checks (like athletics to vault a fence) in the latter case, as obstacles might certainly slow them down as well. Just because you know a potentially faster route doesn't mean you necessarily have the means to take advantage of it.
Simulation is only an approximation. Unless you've determined every event that happens in a city, and every location of every person within that city at every moment, I think it's perfectly fair for the players to ask the DM to let the dice decide instead of forcing them to play through some arbitrary scripted event.