So they head out from Earth toward the sun at about warp 8...

... and it takes them about a minute to get there? The sun is only about 8 light minutes from Earth, so warp 8 is only 8 times the speed of light?

That's not how it worked in the original series. If it had worked that way they never would have gotten anywhere fast, since warp 8 was close to their maximum speed without pushing the limit too hard. At only 8 times the speed of light it would have taken around 6 months just to reach the nearest star from Earth, which is ~4.25 light years away.

In the episode "That Which Survives" they traveled 990.7 light years in about half a day at a stated speed of warp 8.4, which means warp 8.4 is about 723,706 times the speed of light in that episode.