As
@Campbell has said, maintaining consistency of the fiction is pretty much GMing 101. It's not a distinctive technique or particularly well-adapted to one rather than another sort of play experience.
But being relaxed about times and distances isn't about inconsistency. It's about not being needlessly specific. In the particular bit of play that I described, it was established that the castle was an easy ride from the coastal village in the night-time. This certainly does not preclude it being a comfortable walking distance from the coastal village by daylight. Are we talking 2 miles? 5 miles? Even a little further than that? It doesn't matter.
The events that took place while the characters went to the lighthouse included some parleying, a joust with 3 lances but resting between each lance, and then a pitched battle between 100+ soldiers that ended with one side fleeing. How long exactly did all that take? Again, it doesn't matter.
No one at the table had any sense of inconsistency or spoiled verisimilitude.
I don't see consistency as binary. Rather, I see it as a sliding scale. Sure, all playstyles value consistency, but I would argue that exploratory play places a much higher premium on consistency than a heightened-drama playstyle. In exploratory play it's not enough to avoid outright inconsistency, instead a goal of play is to demonstrate the consistency of the setting. This is commonly done by accurate adhereance to a pre-written setting if the exploratory playstyle is accompanied by a DM-as-referee DMing style, but as I described in a previous post I think the requisite heightened awareness of the consistency of the setting can be achieved through other DMing styles as well.
Does this mean that you disagree with
@Ovinomancer's suggestion that 5e wouldn't support the sort of play that I described?
I'm not trying to engender fights here, just trying to get a sense of the analytical and dialetical terrain, and the place of various posters on it.
I haven't seen anyone report experience with running 5e with a playstyle of a DMing style similar to PBtA (admittedly, the distinction between playstyle and DMing style appears to matter less in PBtA where they are so closely intertwined). My experience with PBtA is limited to a single campaign of Urban Shadows, so I don't feel qualified to opine on whether 5e is flexible enough to accommodate that style. (Please also see my response to
@Ovinomancer below.)
5e doesn't support a wide range of playstyles, though. It supports a narrow range of playstyles strongly constrained by "GM adjudicates." It's further constrained under this by quite a lot of "GM says." The rules are open to GM ruling, by design, meaning quite a bit of a game is, mechanically, what the GM says it is.
Really, what we've been arguing here are small differences -- essentially hiw players are expected to interact with GM provided fuction so that the GM can adjudicate according to GM preferences. Yes, it seems big, but if you compare it to other ganes where player have direct control over adding fictional elements in play and limited GM fiat, then, no, it's really not a broad range of play supported by 5e.
And, that's actually really good! You don't want such a malleable game -- it would suck. 5e is good at being D&D, and that's awesome.
Two tables of 5e can have entirely different playstyles and DMing styles and still be successful. For example, my 5e games emphasize player-driven exploratory play where combat difficulty depends mostly on the strategic choices the PCs and their opponents make prior to rolling initiative. At the same time, I run the game in a DM-as-Entertainer style where accurate refereeing with reference to pre-written material has no intrinsic value. I will modify the (unseen parts of the) game world on the fly to control pacing, drama, and increase enjoyability, but my framing and telegraphing of that content is always a neutral adjudication that avoids deliberate stake-setting.
By contrast, assuming I am understanding correctly, many of the posters in this thread DM in a style that values accurately adhering to their pre-written material, but will consciously frame and telepgraph that content as a tool to control pacing, drama, and enjoyability and promote deliberate stake-setting. There has been less discussion of playstyles than DMing styles, but I get the impression that many of those posters favor DM-driven, drama-focused styles where players are tactically reacting to the material as it is being presented (with some difference of opinion on how immediate those reactions and their consequences should be).
So we've already got two almost-inverted combinations of playstyles and DMing styles that 5e apparently works for. Then we've also had a couple posters in this thread who fully support an exploratory playstyle with a classic DM-as-referee DMing style, and 5e works for them too.
These three entirely-different combinations are sufficient variety for me to stand by my assertion that 5e supports a wide variety of playstyles and DMing styles. Sure, as I said to
@pemerton above, I don't know if 5e would support a PBtA play/DM style, and I also don't know how well it would work in playstyles that permit the players to add fictional elements of the game world. But even if 5e won't support those styles, I still feel justified that it supports enough disperate styles to qualify as supporting a wide range.