You either are willfully ignoring material or construing evidence in such a way as to make it a convoluted mess.
It IS a convoluted mess. Has been for years.
That said, the Enterprise CAN be a "few light days away" from Delta Vega- say three light days out when Kirk & Spock have their discussion in the briefing room, travel there at 89% of the speed of light and make it there via time dilation effects in only a day and half or thereabouts shipboard time.
Sure, but they wouldn't have been a few light days away from Delta-Vega
when they crossed the barrier.
And to allude to the earlier discussion, if you're assuming that there is some sort of subspace fuckery going on with impulse drive anyway (which there almost certainly is) then there's no reason to assume the ship would be limited to subluminal velocity under impulse power. That's kind of what the "years away" in Kirk's log entry implies: they could still travel dozens of light years to get to a starbase, but it probably wouldn't take them a whole generation to do it.
3 light days would be a distance of about 540 AUs. If Enterprise flew a brachistochrone trajectory and accelerated at a constant rate of 15 Gs, it would cover that distance in about two and a half weeks. Bump that up to 30Gs constant thrust, and it would cover that distance in almost exactly 12 days. Significantly, the ship's average velocity would be relatively low... OTOH, at the deceleration phase it would achieve a maximum velocity of about 508 times the speed of light.
Now here's where it dovetails back to the original discussion: if a warp factor is a unit of acceleration rather than absolute velocity (and it pretty much has to be) then the travel time to Delta Vega finally makes sense. Enterprise CAN travel interstellar distances under impulse power, and exceeding light speed is trivially easy. But impulse engines have the problem that you also have to decelerate in order to actually reach your destination, an the deceleration curve of impulse engines is far more limited compared to warp engines. Where where an acceleration at warp 8 might be, say, 180,000 Gs, an impulse engine might be able to achieve only a tiny fraction of that even on a good day (30 to 40gs), while a 22nd century impulse engine (like the kind the Valiant had that "wasn't strong enough" to resist an ion storm) would be slower still, closer to 10Gs. That would leave impulse engines to be a high-efficiency short range drive for situations where warp drive might be overkill or inappropriate altogether, such as orbital maneuvers or short interplanetary journeys within tightly packed moons.
For the record: at 980,000G constant acceleration (what I would gather to be about warp 6), the Enterprise would be able to reach Delta Vega from that position in 3.8 hours. At that same acceleration, making a beeline for four days, the Enterprise would cover a distance of about 121 light years. These are the kinds of speeds that seem more consistent with what we're seeing all over TOS; "The Immunity Syndrome" and "That Which Survives" comes to mind as the Enterprise achieving first incredibly high velocities for no obvious reason. The answer is: it's not about velocity, the Enterprise sometimes reaches many tens of thousands of times the speed of light, but it has to SLOW DOWN for the second half of its trip in order to not completely overshoot its destination, and that long slowdown stage accounts for part of the travel time.
If you doubt what I have to say, you can input the values yourself in this handy dandy
time dilation calculator.
Time dilation has never been a factor in Star Trek and probably never will be.