^^ You've hit on something that I feel has long been ignored from "The Cage" (and remember that the scene you're referencing was also seen in "The Menagerie" so it cannot be discounted as never having happened) and something that was reinforced later in TOS: the idea that not too long before the Pike and Kirk era star flight technology was distinctly slower and more crude. Yeah, one could argue it being a matter of degrees, but the subtext in all of it flies in contradiction of what they did in ENT.
Here's a little something I worked out a few years ago for a project I have going off-and-on. I used the formula to calculate the relativistic time dilation effect for various speeds. In this example I used Alpha Centauri as the target destination, but the same formula could be used for any target star. From this you can get a clearer idea of what a crew would experience time dilation at different speeds.
To show it onscreen you could have two clocks. One clock would be set to show the passage of time for the crew and it would look completely normal. The other clock would be set to illustrate what was happening in the objective universe outside of the ship and this clock would be racing crazily until you slowed down and it would gradually slow to eventually be in synch with the first clock.
Now more to the issue at hand. For the Enterprise
to get to Delta Vega within a few days it will have to accelerate to at least .99999c
(99.999% of light) assuming the destination is within about five light years. Even so it would mean some years will have passed objectively. The simple fact is even if you can get to 99.999% of light it still takes a objectively finite amount of time to cross that distance.
Another wrinkle is that Spock say Delta Vega is only a few light days
away. A light day isn't very far. If we use the idea of five days then it's only about 80.352 billion miles or 864 AU. If my math is right then that's a bit more than 1/70th of a light year. You might as well say Delta Vega is just outside the hull and you can toss stones at it in terms of astronomical distances.
So the question: is Spock referring to distance or the time to travel that distance? Actually a light year or a light day is referring to both time and distance, but I suspect Spock is using the term in as such as to give the viewers a spacey way of expressing the time it takes to travel to Delta Vega. In that context then it more strongly supports the idea that Delta Vega could be within five light years away and they can get there under impulse albeit pushing it to 99.99% of light. But that doesn't get you away from the relativistic problem: they'll get to Delta Vega and lose a few years in the process. Well, actually they don't lose those years themselves because they'll age only a few days, but anyone else back home will think the Enterprise
disappeared for a few years.
No matter how you cut it our heroes are in a pickle.
Now there could be a loop hole. If the Enterprise's
impulse engines can generate a subspace field and allow them to get a bit faster than Warp 1 then they can make it to Delta Vega in a few days and be spared any time dilation effects. The one wrinkle to that is Warp 1 can't be = to the speed of light, something I've long suspected in TOS.