You're assuming a mistake has been made, when I suggest they're deliberately using the faster examples I've cited as their baseline - moving the goal posts back to where they were, so to speak.
No, TOS never treated the trip from Earth to Vulcan, or the Klingon border, as nigh-instantaneous. TOS was about a starship exploring
deep space, far from its home port and never returning there. The modern series' effortless commutes from Earth to everywhere else are a complete abandonment of the original intent of Trek. It's not much of a frontier if you get there faster than you can walk to the corner store.
And as I said, we know for a fact that the filmmakers of ST '09 structured the trip to Vulcan to imply it took longer than shown; Abrams wanted it to flow smoothly for the sake of pacing, but subtly acknowledged the underlying reality that it would realistically have to take more than a few moments. McCoy's change of clothing and Kirk's time spent sedated suggest that a few hours at least have elapsed even though the editing implies a quicker pace. This was entirely intentional, as we know for certain from statements made by the filmmakers. So STID's failure to acknowledge the same passage of time in a trip from the Klingon border to Earth contradicts the intentions of the previous film. And thus, yes, I think I can validly call it a mistake, an oversight on the filmmakers' part -- or, if it was an intentional oversight, still a bad and unnecessary idea.
IMHO everything apart from Voyager fits nicely into this framework - no need to ignore lines like "rim of the galaxy" and "centre of the galaxy"
And I've already proven that that's not true, that VGR is merely following the precedent established in TNG and DS9. I cited DS9's "Battle Lines" as establishing specific travel time, but I forgot that the precedent was actually set far earlier in TNG's "The Price." The Barzan wormhole was said to span 70,000 light years (the same distance as the Bajoran wormhole and nearly the same as
Voyager's journey), and Picard said that the travel time at normal warp would be "eighty years or so." There's also "Q Who," in which the ship was thrown 7,000 light-years across space and Data said it would take two years, seven months to reach the nearest starbase. (Not to return to their original position, though, so perhaps that nearest starbase is substantially closer than where they started from.) So it's been consistent throughout TNG, DS9, and VGR that 24th-century warp drive would take years to span thousands of light-years. The specific cited travel times are inconsistent, but all within an order of magnitude of 1000 times the speed of light.
Indeed, the whole concept of DS9 goes out the window if you can get from Earth to Vulcan (canonically 16 light-years) in a couple of minutes, or to the center of the galaxy (c. 27,000 light-years) in less than an hour. At the former speed, Dominion space would be only a few days away, and at the latter it would be only a couple of hours away. In either case, the wormhole would have no value or strategic importance. Obviously the entire series of DS9 depends on that not being the case, on inter-quadrant travel taking decades via conventional warp drive. As do TNG episodes like "Where No One Has Gone Before," "Q Who," "The Price," "The Nth Degree," and "Descent."
So you're absolutely wrong to say that VGR is the only series based on this premise. The same limit on warp velocities applies throughout TNG, DS9, VGR, and obviously ENT where the ships were slower. That's the vast majority of the entire Trek franchise -- 628 episodes/films out of a total 738 to date (counting "The Cage" and both parts of "The Menagerie" separately). That's 85% of canon.
When the whole point of the story is that they're leaving the galaxy or travelling to it's centre, it seems rather silly to pretend they didn't know where they really were.
As stated, the galactic center is 27,000 +/- 1000 light-years from Earth. The thin disk of the galaxy has no real definable edge, of course, but it's estimated to be roughly 3300 ly thick, meaning the nearest face would be about 1600 ly away. Although it's irregular enough that there's no way to define that exactly. Of course the whole idea of an "energy barrier at the edge of the galaxy" is complete nonsense.